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Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2013 Article ID 491628 18 pageshttpdxdoiorg1011552013491628
Research ArticleThe Beta Generalized Half-Normal Distribution New Properties
Gauss M Cordeiro1 Rodrigo R Pescim2 Edwin M M Ortega2 and Clarice G B Demeacutetrio2
1 Departamento de Estatıstica Universidade Federal de Pernambuco 50749-540 Recife PE Brazil2 Departamento de Ciencias Exatas Universidade de Sao Paulo ESALQ 13418-900 Piracicaba SP Brazil
Correspondence should be addressed to Rodrigo R Pescim rrpescimgmailcom
Received 3 May 2013 Revised 1 November 2013 Accepted 6 November 2013
Academic Editor Ricardas Zitikis
Copyright copy 2013 Gauss M Cordeiro et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We study some mathematical properties of the beta generalized half-normal distribution recently proposed by Pescim et al (2010)This model is quite flexible for analyzing positive real data since it contains as special models the half-normal exponentiated half-normal and generalized half-normal distributions We provide a useful power series for the quantile function Some new explicitexpressions are derived for the mean deviations Bonferroni and Lorenz curves reliability and entropy We demonstrate that thedensity function of the beta generalized half-normal order statistics can be expressed as a mixture of generalized half-normaldensities We obtain two closed-form expressions for their moments and other statistical measures The method of maximumlikelihood is used to estimate themodel parameters censored dataThe beta generalized half-normalmodel ismodified to cope withlong-term survivors may be present in the dataThe usefulness of this distribution is illustrated in the analysis of four real data sets
1 Introduction
Cooray and Ananda [1] pioneered the generalized half-normal (GHN) distribution with shape parameter 120572 gt 0 andscale parameter 120579 gt 0 defined by the cumulative distributionfunction (cdf)
119866120572120579
(119909) = 2Φ[(
119909
120579
)
120572
] minus 1 = erf [(119909120579)120572
radic2
] (1)
where the standard normal cdf Φ(119909) and the error functionerf(119909) are given by
Φ (119909) =
1
2
[1 + erf ( 119909
radic2
)] erf (119909) = 2
radic120587
int
119909
0
119890minus1199052
119889119905
(2)
Following an idea due to Eugene et al [2] Pescim etal [3] proposed the beta generalized half-normal (BGHN)distributionwhich seems to be superior over theGHNmodelfor some applications The justification for the practicabilityof this model is based on the fatigue crack growth undervariable stress or cyclic load In this paper we study severalmathematical properties of the BGHN model with the hope
that it will attract wider applications in reliability engineeringand in other areas of research
The four-parameter BGHN cdf is defined from (1) by (for119909 gt 0)
119865 (119909) = 1198682Φ[(119909120579)
120572]minus1
(119886 119887)
=
1
119861 (119886 119887)
int
2Φ[(119909120579)120572]minus1
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908
(3)
where 119861(119886 119887) = [Γ(119886)Γ(119887)]Γ(119886 + 119887) is the beta function119868119910(119886 119887) = 119861(119886 119887)
minus1
int
119910
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908 is the incompletebeta function ratio and 119886 gt 0 and 119887 gt 0 are two additionalshape parameters
The probability density function (pdf) and the hazard ratefunction (hrf) corresponding to (3) are
119891 (119909) =
2119887minus1
radic2120587 (120572119909) (119909120579)120572
119890minus(12)(119909120579)
2120572
119861 (119886 119887)
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909
120579
)
120572
]
119887minus1
(4)
2 Journal of Probability and Statistics
ℎ (119909) = (2119887minus1
radic2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909
120579
)
120572
]
119887minus1
)
times (119861 (119886 119887) 1 minus 1198682Φ[(119909120579)
120572]minus1
(119886 119887))minus1
(5)
respectively Hereafter a random variable with pdf (4) isdenoted by119883 sim BGHN (120572 120579 119886 119887)
Pescim et al [3] demonstrated that the cdf and pdf of119883 can be expressed as infinite power series of the GHNcumulative distribution Here all expansions in power seriesare around the point zero If 119887 gt 0 is a real noninteger we canexpand the binomial term in (3) to obtain
119865 (119909) =
infin
sum
119895=0
1199081198952Φ[(
119909
120579
)
120572
] minus 1
119886+119895
(6)
where 119908119895= 119908
119895(119886 119887) = (minus1)
119895
119861(119886 119887)minus1
(119886 + 119895)minus1
(119887minus1
119895)
The pdf corresponding to (6) can be expressed as
119891 (119909) = radic2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times
infin
sum
119895=0
1199081198952Φ[(
119909
120579
)
120572
] minus 1
119886+119895minus1
(7)
If 119886 gt 0 is an integer (7) provides the BGHN density func-tion as an infinite power series of the GHN cumulative dis-tribution If 119887 is an integer the index 119895 in the previous sumstops at 119887 minus 1 Otherwise if 119886 is a real noninteger we canexpand 2Φ[(119909120579)120572] minus 1119886+119895minus1 as follows
2Φ[(
119909
120579
)
120572
] minus 1
119886+119895minus1
=
infin
sum
119903=0
119904119903(119886 + 119895 minus 1) 2Φ [(
119909
120579
)
120572
] minus 1
119903
(8)
where
119904119903(119886 + 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886 + 119895
(
119886 + 119895 minus 1
119896)(
119896
119903) (9)
Hence
119891 (119909) = radic2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572infin
sum
119903=0
119887119903erf [(119909120579)
120572
radic2
]
119903
(10)
whose coefficients are
119887119903= 119887
119903(119886 119887) =
1
119861 (119886 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886 + 119895 minus 1) (11)
TheBGHNdensity function (4) allows for greater flexibil-ity of its tails and can be widely applied inmany areas of engi-neering and biology We study the mathematical properties
of this distribution because it extends some important dis-tributions previously considered in the literature In fact theGHN model with parameters 120572 and 120579 is clearly an import-ant special case for 119886 = 119887 = 1 with a continuous crossovertowards models with different shapes for example a par-ticular combination of skewness and kurtosis The BGHNdistribution also contains the exponentiated generalized half-normal (EGHN) and half-normal (HN) distributions assubmodels for 119887 = 1 and 119886 = 119887 = 120572 = 1 respectively More-over while the transformation (3) is not analytically tractablein the general case the formulas related to the BGHNdistribution turn outmanageable andwith the use ofmoderncomputer resources with analytic and numerical capabilitiesmay turn into adequate tools comprising the arsenal ofapplied statisticians
The paper is outlined as follows We derive an expansionfor the quantile function in Section 2 Some statistical mea-sures for the BGHN distribution such as moments generat-ing function mean deviations Renyi entropy and reliabilityare studied in Section 3 The computational issues relatingto the infinite series for structural properties of the BGHNdistribution are discussed in Section 4 In Section 5 we derivealgebraic expressions for the moments moment generatingfunction (mgf) mean deviations and Renyi entropy for theorder statistics The estimation by maximum likelihoodmdashincluding the case of censoringmdashis presented in Section 6In Section 7 we propose a BGHN mixture model for sur-vival data with long-term survivors Section 8 illustrates theimportance of the BGHN distribution applied to four realdata sets Finally concluding remarks are given in Section 9
2 Power Series for the Quantile Function
Power series methods are at the heart of many aspects ofapplied mathematics and statistics Quantile functions arein widespread use in probability distributions and generalstatistics and often find representations in terms of powerseriesThe quantile function for a probability distribution hasmany uses in both the theory and application of probabilityIt may be used to generate values of a random variable having119865(119909) as its distributions functionThis fact serves as the basisof a method for simulating a sample from an arbitrary distri-bution with the aid of a uniform random number generator
The quantile function of 119883 say 119909 = 119876BGHN(119906) = 119865minus1
(119906)can be obtained by inverting the cumulative function (3)Now we provide a power series expansion for 119876BGHN(119906)that can be useful to determine some mathematical mea-sures of the BGHN distribution First an expansion forthe inverse of the incomplete beta function 119868
119909(119886 119887) = 119906
can be found in wolfram website (httpfunctionswolframcom062306000401) as
119911 = 119876119861(119906) =
infin
sum
119894=0
119889119894119906119894119886
(12)
where119876119861(119906) is the beta quantile function 119889
119894= 119890
119894[119886119861(119886 119887)]
1119886
for 119886 gt 0 and 1198900= 0 119890
1= 1 119890
2= (119887 minus 1)(119886 + 1)
Journal of Probability and Statistics 3
The coefficients 1198901015840119894s for 119894 ge 2 can be obtained from a cubic
recursion of the form
119890119894=
1
[1198942+ (120572 minus 2) 119894 + (1 minus 120572)]
times (1 minus 1205751198942)
119894minus1
sum
119903=2
119890119903119890119894+1minus119903
times [119903 (1 minus 119886) (119894 minus 119903) minus 119903 (119903 minus 1)]
+
119894minus1
sum
119903=1
119894minus119903
sum
119904=1
119890119903119890119904119890119894+1minus119903minus119904
times [119903 (119903 minus 119886) + 119904 (119886 + 119887 minus 2) (119894 + 1 minus 119903 minus 119904)]
(13)
where 1205751198942= 1 if 119894 = 2 and 120575
1198942= 0 if 119894 = 2 In the last equation
we note that the quadratic term only contributes for 119894 ge 3Following Steinbrecher [4] the quantile function of the
standard normal distribution say 119876119873(119906) = Φ
minus1
(119906) can beexpanded as
119876119873(119906) = Φ
minus1
(119906) =
infin
sum
119896=0
1198871198961205772119896+1
(14)
where 120577 = radic2120587(119906 minus 12) and the 1198871015840119896s can be calculated from
1198870= 1 and
119887119896+1
=
1
2 (2119896 + 3)
119896
sum
119903=0
(2119903 + 1) (2119896 minus 2119903 + 1) 119887119903119887119896minus119903
(119903 + 1) (2119903 + 1)
(15)
Here 1198871= 16 119887
2= 7120 119887
3= 1277560
The function 119876119873(119906) can be expressed as a power series
given by
119876119873(119906) =
infin
sum
119895=0
119891119895119906119895
(16)
where 119891119895= sum
infin
119896=119895(minus12)
119896minus119895
(119896
119895) 119892
119896and the quantities 119892
119896are
defined from the coefficients in (14) by 119892119896
= 0 for 119896 =
0 2 4 and 119892119896= (2120587)
1198962
119887(119896minus1)2
for 119896 = 1 3 5 By inverting the normal cumulative function in 119911 =
2Φ[(119909120579)120572
] minus 1 and using (16) we can express the quantilefunction of119883 in terms of 119911 as
119909 = 120579[119876119873(
119911 + 1
2
)]
1120572
= 120579[
[
infin
sum
119895=0
119891119895(
119911 + 1
2
)
119895
]
]
1120572
= 120579[
[
infin
sum
119895=0
119895
sum
119901=0
119891119895
2119895(
119895
119901) 119911
119901]
]
1120572
(17)
Replacingsuminfin
119895=0sum119895
119901=0bysuminfin
119901=0suminfin
119895=119901in last equation we obtain
119909 = 120579 (
infin
sum
119901=0
ℎ119901119911119901
)
1120572
(18)
where ℎ119901= sum
infin
119895=1199012minus119895
119891119895(119895
119901) Using the same steps of (8) we
can write
(
infin
sum
119901=0
ℎ119901119911119901
)
1120572
=
infin
sum
119903=0
119904119903(120572
minus1
)(
infin
sum
119901=0
ℎ119901119911119901
)
119903
(19)
where 119904119903(120572
minus1
) = suminfin
119895=119903(minus1)
119903+119895
(120572minus1
119895) (
119895
119903)
We use throughout an equation of Gradshteyn andRyzhik [5] for a power series raised to a positive integer 119895
(
infin
sum
119894=0
119886119894119909119894
)
119895
=
infin
sum
119894=0
119888119895119894119909119894
(20)
where the coefficients 119888119895119894(for 119894 = 1 2 ) are easily obtained
from the recurrence equation
119888119895119894= (119894119886
0)minus1
119894
sum
119898=1
[119898 (119895 + 1) minus 119894] 119886119898119888119895119894minus119898
(21)
and 1198881198950
= 119886119895
0 From (19) and (20) we have
119909 =
infin
sum
119901=0
ℎ⋆
119901119911119901
(22)
where ℎ⋆
119901= 120579sum
infin
119903=0119904119903(120572
minus1
) ℎ119903119901
for 119901 ge 0 ℎ119903119901
=
(119901ℎ0)minus1
sum119901
119898=1[119898(119903 + 1) minus 119901] ℎ
119898ℎ119903119901minus119898
for 119901 ge 1 and ℎ1199030
= ℎ119903
0
By inserting (12) in (22) we obtain
119909 = 119876BGHN (119906) =infin
sum
119901=0
ℎ⋆
119901(
infin
sum
119894=0
119889119894119906119894119886
)
119901
(23)
From (20) it follows (suminfin
119894=0119889119894119906119894119886
)
119901
= suminfin
119894=0119889119901119894119906119894119886 where the
quantities 119889119901119894
are determined recursively from 1198890119894= 119889
119894
0and
119889119901119894
= (1198941198890)minus1
sum119894
119898=1[119898 (119901 + 1) minus 119894]119889
119898119889119901119894minus119898
(for 119894 = 1 2 )Finally we obtain
119909 = 119876BGHN (119906) =infin
sum
119894=0
V119894119906119894119886
(24)
where V119894= sum
infin
119901=0ℎ⋆
119901119889119901119894 Equation (24) gives the BGHN
quantile function as a power series and represents the mainresult of this section
3 BGHN Properties
31Moments andGenerating Function Here we provide newexpressions for the moments and mgf of 119883 based upon thepower series for its quantile function as alternative resultsform those obtained by Pescim et al [3]
We can write from (24)
1205831015840
119904(120572 120579 119886 119887) = 119864 (119883
119904
) = int
infin
0
119909119904
119891 (119909) 119889119909
= int
1
0
(
infin
sum
119894=0
V119894119906119894119886
)
119904
119889119906
(25)
4 Journal of Probability and Statistics
and then using (20) we obtain
1205831015840
119904(120572 120579 119886 119887) = 119886
infin
sum
119894=0
V119904119894
(119894 + 119886)
(26)
where the quantities V119904119894(for 119894 = 1 2 ) are easily determined
from the recurrence equation
V119904119894= (119894V
0)minus1
119894
sum
119898=1
[119898 (119904 + 1) minus 119894] V119898V119904119894minus119898
(27)
with V1199040
= V1199040
We now provide a new alternative representation for themgf of 119883 say 119872
120572120579119886119887(119905) = 119864(119890
119905119883
) based on the quantilepower series (24) We can write
119872120572120579119886119887
(119905) = int
infin
0
119890119905119909
119891 (119909) 119889119909
= int
1
0
exp[119905(infin
sum
119894=0
V119894119906119894119886
)]119889119906
(28)
We expand the exponential function and use the same algebrathat leads to (26)
119872120572120579119886119887
(119905) =
infin
sum
119903=0
119905119903
119903
int
1
0
(
infin
sum
119894=0
V119894119906119894119886
)
119903
119889119906
=
infin
sum
119903119894=0
119905119903V119903119894
119903
int
1
0
119906119894119886
119889119906
(29)
and then
119872120572120579119886119887
(119905) = 119886
infin
sum
119903119894=0
119905119903V119903119894
(119886 + 119894) 119903
(30)
Equations (26) and (30) are the main results of this section
32 Mean andMedian Deviations The amount of scatter in apopulation is evidently measured to some extent by the meandeviations in relation to the mean and the median defined by
1205751(119883) = int
infin
0
10038161003816100381610038161003816119909 minus 120583
1015840
1
10038161003816100381610038161003816119891 (119909) 119889119909
1205752(119883) = int
infin
0
|119909 minus119872|119891 (119909) 119889119909
(31)
respectively where 1205831015840
1= 119864(119883) and 119872 = Median(119883)
denotes the median Here119872 is calculated as the solution ofthe nonlinear equation 119868
2Φ[(119872120579)120572]minus1(119886 119887) = 12 We define
119879(119902) = int
infin
119902
119909119891(119909)119889119909 which is determined below The mea-sures 120575
1(119883) and 120575
2(119883) can be written in terms of 1205831015840
1and 119879(119902)
as
1205751(119883) = 2120583
1015840
1119865 (120583
1015840
1) minus 2120583
1015840
1+ 2119879 (120583
1015840
1)
1205752(119883) = 2119879 (119872) minus 120583
1015840
1
(32)
For more details see Paranaıba et al [6] Clearly 119865(119872) and119865(120583
1015840
1) are determined from (3) From (10) we have
119879 (119902)
= 120572radic2
120587
infin
sum
119903=0
119887119903int
infin
119902
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
erf [(119909120579)120572
radic2
]
119903
119889119909
(33)
Setting 119906 = (119909120579)120572 in the last equation gives
119879 (119902)
= 120579radic2
120587
infin
sum
119903=0
119887119903int
infin
(119902120579)1205721199061120572
119890minus11990622
[erf ( 119906
radic2
)]
119903
119889119906
(34)
Using the power series for the error function erf(119909) =
(2radic120587)suminfin
119898=0((minus1)
119898
1199092119898+1
(2119898 + 1)119898) (see eg [7]) weobtain after some algebra
119879 (119902) = 120579radic2
120587
infin
sum
119903=0
119887119903(
2
radic120587
)
119903
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119903=0
(minus1)1198981+sdotsdotsdot+119898
119903
(21198981+ 1) sdot sdot sdot (2119898
119903+ 1)119898
1 sdot sdot sdot 119898
119903
times Γ [(1198981+ sdot sdot sdot + 119898
119903+
119903
2
+
1
2120572
+
1
2
)
1
2
(
119902
120579
)
120572
]
(35)
where Γ(119901 119909) = int
infin
119909
V119901minus1119890minusV119889V denotes the complementaryincomplete gamma function for 119901 gt 0 The measures 120575
1(119883)
and 1205752(119883) are immediately calculated from (35)
Bonferroni and Lorenz curves have applications not onlyin economics to study income and poverty but also inother fields such as reliability demography insurance andmedicine They are defined by
119861 (120588) =
1
1205881205831015840
1
int
119902
0
119909119891 (119909) 119889119909
119871 (120588) =
1
1205831015840
1
int
119902
0
119909119891 (119909) 119889119909
(36)
respectively where 119902 = 119876BGHN(120588) = 119876119861(]) and ] =
2Φ[(120588120579)120572
] minus 1 (Section 2) for a given probability 120588 Fromint
119902
0
119909119891(119909)119889119909 = 1205831015840
1minus 119879(120588) we obtain 119861(120588) = 120588
minus1
[1 minus 119879(120588)1205831015840
1]
and 119871(120588) = 1 minus 119879(120588)1205831015840
1
33 Renyi Entropy The Renyi information of order 120585 for acontinuous random variable with density function 119891(119909) isdefined as
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (37)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 Applicationsof the Renyi entropy can be found in several areas suchas physics information theory and engineering to describe
Journal of Probability and Statistics 5
many nonlinear dynamical or chaotic systems [8] and instatistics as certain appropriately scaled test statistics (relativeRenyi information) for testing hypotheses in parametricmodels [9] Renyi [10] generalized the concept of informationtheory which allows for different averaging of probabilitiesvia 120585
For the BGHN distribution (4) the Renyi entropy isdefined by
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (38)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 From (4) we have
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
120585(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120585(119887minus1)
119889119909
(39)
For |119911| lt 1 and 119887 is a real noninteger the power series holds
(1 minus 119911)119887minus1
=
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119911
119895
(40)
where the binomial coefficient is defined for any real Using(40) in (39) twice 119868(120585) can be expressed as
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times
infin
sum
119895119896=0
(minus1)119895+119896
2119895
(
120585 (119886 minus 1) + 1
119895)
times (
120585 (119887 minus 1) + 119895 + 1
119896)
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
Φ[(
119909
120579
)
120572
]
119896
119889119909
(41)
Substituting Φ(119909) by the error function and setting 119906 =
(119909120579)120572 119868(120585) reduces to
119868 (120585) = 120572120585minus1
1205791minus120585
2120585(119887minus1)
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times int
infin
0
119906((120585(120572minus1)+1)120572)minus1
119890minus(1205852)119906
2
[erf ( 119906
radic2
)]
119897
119889119906
(42)
Following similar algebra that lead to (35) we obtain
119868 (120585) = 120572120585minus1
1205791minus120585
2[120585(119887minus1)+(120585(120572minus1)+1)2120572minus1]
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
(21198981+ 1) sdot sdot sdot (2119898
119897+ 1)119898
1 sdot sdot sdot 119898
119897
times 120585minus[1198981+sdotsdotsdot+119898
119897+1198972+(120585(120572minus1)+1)2120572]
times Γ(1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
)
(43)
Finally the Renyi entropy reduces to
120485119877(120585) = (1 minus 120585)
minus1
times
(120585 minus 1) log (120572) + (1 minus 120585) log (120579)
+ [120585 (119887 minus 1) +
120585 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120585 log(radic 2
120587
) + log[
[
infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)]
]
+ log[infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
times ((21198981+ 1) sdot sdot sdot (2119898
119897+ 1)
times 1198981 sdot sdot sdot 119898
119897)minus1
]
minus [1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
] log (120585)
+ log [Γ(1198981+sdot sdot sdot+119898
119897+
119897
2
+
120585 (120572minus1) + 1
2120572
)]
(44)
34 Reliability In the context of reliability the stress-strengthmodel describes the life of a component which has a randomstrength 119883
1that is subjected to a random stress 119883
2 The
component fails at the instant that the stress applied toit exceeds the strength and the component will functionsatisfactorily whenever 119883
1gt 119883
2 Hence 119877 = Pr(119883
2lt
1198831) is a measure of component reliability Here we derive
119877 when 1198831and 119883
2have independent BGHN(120572 120579 119886
1 1198871)
and BGHN(120572 120579 1198862 1198872) distributions with the same shape
parameters 120572 and 120579 The reliability 119877 becomes
119877 = int
infin
0
1198911(119909) 119865
2(119909) 119889119909 (45)
6 Journal of Probability and Statistics
where the cdf of 1198832and the density of 119883
1are obtained from
(6) and (10) as
1198652(119909) =
infin
sum
119895=0
119908119895(119886
2 1198872) 2Φ [(
119909
120579
)
120572
] minus 1
1198862+119895
1198911(119909) = radic
2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times
infin
sum
119903=0
119887119903(119886
1 1198871) 2Φ [(
119909
120579
)
120572
] minus 1
119903
(46)
respectively where
119908119895(119886
2 1198872) =
(minus1)119895
119861 (1198862 1198872)
(
1198872minus 1
119895)
119887119903(119886
1 1198871) =
infin
sum
119895=0
(minus1)119895
119861 (1198861 1198871)
(
1198871minus 1
119895) 119904
119903(119886
1+ 119895 minus 1)
(47)
refer to1198832and119883
1 respectively Hence
119877 = 120572radic2
120587
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
times int
infin
0
119909minus1
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
1198862+119895+119903
119889119909
(48)
Setting 119906 = 2Φ[(119909120579)120572
] minus 1 in the last integral the reli-ability of119883 reduces to
119877 =
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
1198862+ 119895 + 119903 + 1
(49)
4 Computational Issues
Here we show the practical use of (24) (26) (30) (35) and(49) If we truncate these infinite power series we have asimple way to compute the quantile function moments mgfmean deviations and reliability of the BGHN distributionThe question is how large the truncation limit should be
We now provide evidence that each infinite summationcan be truncated at twenty to yield sufficient accuracy Let119863
1
denote the absolute difference between the integrated version
1
119861 (119886 119887)
int
2Φ[(119909120579)120572]minus1
0
119905119886minus1
(1 minus 119905)119887minus1
119889119905 = 119880 (50)
where 119880 sim 119880(0 1) is a uniform variate on the unit intervaland the truncated version of (24) averaged over (all with step001) 119909 = 001 5 119886 = 001 10 119887 = 001 10
120572 = 001 10 and 120579 = 001 10 Let 1198632denote the
absolute difference between integrated version
1205831015840
119904=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
119909119904
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(51)
and the truncated version of (26) averaged over 119909 =
001 5 119886 = 001 10 119887 = 001 10120572 = 001 10
and 120579 = 001 10 Let 1198633denote the absolute difference
between the truncated version of (30) and the integratedversion
119872120572120579119886119887
(119905)
=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
(
120572
119909
)(
119909
120579
)
120572
exp [119905119909 minus 1
2
(
119909
120579
)
2120572
]
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(52)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198634denote
the absolute difference between the truncated version of (35)and the integrated version
119879 (119902) =
120572 2119887minus1
119861 (119886 119887)
radic2
120587
int
119902
0
(
119909
120579
)
120572
times exp [minus12
(
119909
120579
)
2120572
] 2Φ[(
119909
120579
)
120572
]minus1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(53)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198635denote
the absolute difference between the truncated version of (49)and the integrated version (45) averaged over 119909 = 001 5119886 = 001 10 119887 = 001 10 120572 = 001 10 and 120579 =
001 10We obtain the following estimates after extensive compu-
tations1198631= 231times10
minus201198632= 847times10
minus181198633= 122times10
minus211198634= 151 times 10
minus22 and1198635= 941 times 10
minus19 These estimates aresmall enough to suggest that the truncated versions of (24)(26) (30) (35) and (49) are reasonable practical use
It would be important to verify that each (untruncated)infinite series (such as (24) (26) (30) (35) and (49)) is con-vergent and provide valid values for its arguments Howeverthis will be a difficult mathematical problem that could beinvestigated in future work
Journal of Probability and Statistics 7
5 Properties of the BGHN Order Statistics
Order statistics have been used in a wide range of prob-lems including robust statistical estimation and detectionof outliers characterization of probability distributions andgoodness-of-fit tests entropy estimation analysis of censoredsamples reliability analysis quality control and strength ofmaterials
51 Mixture Form Suppose 1198831 119883
119899is a random sample
of size 119899 from a continuous distribution and let1198831119899
lt 1198832119899
lt
sdot sdot sdot lt 119883119899119899
denote the corresponding order statistics Therehas been a large amount of work relating tomoments of orderstatistics119883
119894119899 See Arnold et al [11] David and Nagaraja [12]
and Ahsanullah and Nevzorov [13] for excellent accounts Itis well known that the density function of119883
119894119899is given by
119891119894119899(119909) =
119891 (119909)
119861 (119894 119899 minus 119894 + 1)
119865(119909)119894minus1
[1 minus 119865 (119909)]119899minus119894
(54)
For the BGHN distribution Pescim et al [3] obtained
119891119894119899(119909) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119891119896119872119896(119909) (55)
where 119872119896denotes a sequence (119898
1 119898
119894+119896minus1) of 119894 + 119896 minus 1
nonnegative integers 119891119896119872119896
(119909) denotes the BGHN(120572 120579 (119894 +119896)119886 + sum
119894+119896minus1
119895=1119898119895 119887) density function defined under 119872
119896and
the constants 120578119896119872119896
are given by
120578119896119872119896
=
(minus1)119896+sum119894+119896minus1
119895=1119898119895(119899minus119894
119896) 119861 (119886 (119894 + 119896)+sum
119894+119896minus1
119895=1119898119895 119887) Γ(119887)
119894+119896minus1
119861(119886 119887)119894+119896
119861 (119894 119899minus 119894+1)prod119894+119896minus1
119895=1Γ (119887minus119898
119895)119898
119895 (119886+119898
119895)
(56)
The quantities 120578119896119872119896
are easily obtained given 119896 and asequence 119872
119896of indices 119898
1 119898
119894+119896minus1 The sums in (55)
extend over all (119894 + 119896)-tuples (1198961198981 119898
119894+119896minus1) and can be
implementable on a computer If 119887 gt 0 is an integer (55)holds but the indices 119898
1 119898
119894+119896minus1vary from zero to 119887 minus 1
Equation (55) reveals that the density function of the BGHNorder statistics can be expressed as an infinite mixture ofBGHN densities
52 Moments of Order Statistics Themoments of the BGHNorder statistics can be written directly in terms of themoments of BGHNdistributions from themixture form (55)We have
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119864 (119883119904
119894119896) (57)
where119883119894119896sim BGHN(120572 120579 (119894 + 119896)119886 + sum119894+119896minus1
119895=1119898119895 119887) and 119864(119883119904
119894119896)
can be determined from (26) Inserting (26) in (57) andchanging indices we can write
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
infin
sum
119902=0
119886⋆V
119904119902
(119902 + 119886⋆)
(58)
where
119886⋆
= (119894 + 119896) 119886 +
119894+119896minus1
sum
119895=1
119898119895 (59)
The moments 119864(119883119904
119894119899) can be determined based on the
explicit sum (58) using the software Mathematica They arealso computed from (55) by numerical integration using thestatistical software R [14] The values from both techniquesare usually close wheninfin is replaced by a moderate numbersuch as 20 in (58) For some values 119886 = 2 119887 = 3 and 119899 = 5Table 1 lists the numerical values for the moments 119864(119883119904
119894119899)
(119904 = 1 4 119894 = 1 5) and for the variance skewness andkurtosis
53 Generating Function The mgf of the BGHN orderstatistics can be written directly in terms of the BGHNmgf rsquosfrom the mixture form (55) and (30) We obtain
119872119894119899(119905) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119872120572120579119886⋆119887(119905) (60)
where119872120572120579119886⋆119887(119905) is the mgf of the BGHN(120572 120579 119886⋆ 119887) distri-
bution obtained from (30)
54 Mean Deviations The mean deviations of the orderstatistics in relation to the mean and the median are definedby
1205751(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 120583
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
1205752(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 119872
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
(61)
respectively where 120583119894= 119864(119883
119894119899) and 119872
119894= Median(119883
119894119899)
denotes the median Here 119872119894is obtained as the solution of
the nonlinear equation119899
sum
119903=119894
(
119899
119903) 119868
2Φ[(119872119894120579)120572]minus1
(119886 119887)
119903
times 1 minus 1198682Φ[(119872
119894120579)120572]minus1
(119886 119887)
119899minus119903
=
1
2
(62)
The measures 1205751(119883
119894119899) and 120575
2(119883
119894119899) follow from
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(63)
where 119869119894(119902) = int
infin
119902
119909119891119894119899(119909)119889119909 Using (55) we have
119869119894(119902) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119894119872119896
119879 (119902) (64)
where 119879(119902) is obtained from (35) using 119886⋆ given by (59) and
119887⋆
119903= 119887
119903(119886
⋆
119887) =
1
119861 (119886⋆ 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886
⋆
+ 119895 minus 1)
(65)
8 Journal of Probability and Statistics
Table 1 Moments of the BGHN order statistics for 120579 = 85 120572 = 3 119886 = 2 and 119887 = 3
Order statistics rarr 1198831 5
1198832 5
1198833 5
1198834 5
1198835 5
119904 = 1 267942 581946 473975 171571 023289119904 = 2 1810007 3931172 3201807 1159006 157328119904 = 3 1278683 2777184 2261923 8187821 1111451119904 = 4 938314 2037933 1659828 6008327 8155966Variance 1092078 544561 955281 864636 151904Skewness 057767 minus113596 minus054601 127135 536291Kurtosis 161751 406406 189737 291095 3107642
where
119904119903(119886
⋆
+ 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886⋆+ 119895
(
119886⋆
+ 119895 minus 1
119896)(
119896
119903) (66)
Bonferroni and Lorenz curves of the order statistics aregiven by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(67)
respectively where 119902 = 119865minus1
119894119899(120588) for a given probability 120588 From
int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus 119869
119894(120588) we obtain
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(68)
55 Renyi Entropy The Renyi entropy of the order statisticsis defined by
120485119877(120582) =
1
1 minus 120582
log [119867 (120582)] (69)
where 119867(120582) = int119891120582
119894119899(119909)119889119909 120582 gt 0 and 120582 = 1 From (54) it
follows that
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
int
infin
0
119891120582
(119909)
times [119865 (119909)]120582(119894minus1)
[1minus119865 (119909)]120582(119899minus119894)
119889119909
(70)
Using (40) in (70) we obtain
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
infin
sum
1198961=0
(minus1)1198961(
120582 (119894 minus 1)
1198961
)
times int
infin
0
119891120582
(119909) [119865 (119909)]120582(119894minus1)+119896
1119889119909
(71)
For 120582 gt 0 and 120582 = 1 we can expand [119865(119909)]120582(119894minus1)+1198961 as
119865(119909)120582(119894minus1)+119896
1= 1 minus [1 minus 119865 (119909)]
120582(119894minus1)+1198961
=
infin
sum
1199011=0
(minus1)1199011(
120582 (119894 minus 1) + 1198961+ 1
1199011
) [1 minus 119865 (119909)]1199011
(72)
and then
119865(119909)120582(119894minus1)+119896
1=
infin
sum
1199011=0
1199011
sum
1198971=0
(minus1)1199011+1198971
times (
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)119865(119909)1198971
(73)
Hence from (70) we can write
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
(minus1)1198961+1199011+1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
times int
infin
0
119891120582
(119909) 119865(119909)1198971119889119909
(74)
By replacing 119891(119909) and 119865(119909) by (10) and (16) given by Pescimet al [3] we obtain
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
[Γ (119887)]1198971(minus1)
1198961+1199011+1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Probability and Statistics
ℎ (119909) = (2119887minus1
radic2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909
120579
)
120572
]
119887minus1
)
times (119861 (119886 119887) 1 minus 1198682Φ[(119909120579)
120572]minus1
(119886 119887))minus1
(5)
respectively Hereafter a random variable with pdf (4) isdenoted by119883 sim BGHN (120572 120579 119886 119887)
Pescim et al [3] demonstrated that the cdf and pdf of119883 can be expressed as infinite power series of the GHNcumulative distribution Here all expansions in power seriesare around the point zero If 119887 gt 0 is a real noninteger we canexpand the binomial term in (3) to obtain
119865 (119909) =
infin
sum
119895=0
1199081198952Φ[(
119909
120579
)
120572
] minus 1
119886+119895
(6)
where 119908119895= 119908
119895(119886 119887) = (minus1)
119895
119861(119886 119887)minus1
(119886 + 119895)minus1
(119887minus1
119895)
The pdf corresponding to (6) can be expressed as
119891 (119909) = radic2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times
infin
sum
119895=0
1199081198952Φ[(
119909
120579
)
120572
] minus 1
119886+119895minus1
(7)
If 119886 gt 0 is an integer (7) provides the BGHN density func-tion as an infinite power series of the GHN cumulative dis-tribution If 119887 is an integer the index 119895 in the previous sumstops at 119887 minus 1 Otherwise if 119886 is a real noninteger we canexpand 2Φ[(119909120579)120572] minus 1119886+119895minus1 as follows
2Φ[(
119909
120579
)
120572
] minus 1
119886+119895minus1
=
infin
sum
119903=0
119904119903(119886 + 119895 minus 1) 2Φ [(
119909
120579
)
120572
] minus 1
119903
(8)
where
119904119903(119886 + 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886 + 119895
(
119886 + 119895 minus 1
119896)(
119896
119903) (9)
Hence
119891 (119909) = radic2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572infin
sum
119903=0
119887119903erf [(119909120579)
120572
radic2
]
119903
(10)
whose coefficients are
119887119903= 119887
119903(119886 119887) =
1
119861 (119886 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886 + 119895 minus 1) (11)
TheBGHNdensity function (4) allows for greater flexibil-ity of its tails and can be widely applied inmany areas of engi-neering and biology We study the mathematical properties
of this distribution because it extends some important dis-tributions previously considered in the literature In fact theGHN model with parameters 120572 and 120579 is clearly an import-ant special case for 119886 = 119887 = 1 with a continuous crossovertowards models with different shapes for example a par-ticular combination of skewness and kurtosis The BGHNdistribution also contains the exponentiated generalized half-normal (EGHN) and half-normal (HN) distributions assubmodels for 119887 = 1 and 119886 = 119887 = 120572 = 1 respectively More-over while the transformation (3) is not analytically tractablein the general case the formulas related to the BGHNdistribution turn outmanageable andwith the use ofmoderncomputer resources with analytic and numerical capabilitiesmay turn into adequate tools comprising the arsenal ofapplied statisticians
The paper is outlined as follows We derive an expansionfor the quantile function in Section 2 Some statistical mea-sures for the BGHN distribution such as moments generat-ing function mean deviations Renyi entropy and reliabilityare studied in Section 3 The computational issues relatingto the infinite series for structural properties of the BGHNdistribution are discussed in Section 4 In Section 5 we derivealgebraic expressions for the moments moment generatingfunction (mgf) mean deviations and Renyi entropy for theorder statistics The estimation by maximum likelihoodmdashincluding the case of censoringmdashis presented in Section 6In Section 7 we propose a BGHN mixture model for sur-vival data with long-term survivors Section 8 illustrates theimportance of the BGHN distribution applied to four realdata sets Finally concluding remarks are given in Section 9
2 Power Series for the Quantile Function
Power series methods are at the heart of many aspects ofapplied mathematics and statistics Quantile functions arein widespread use in probability distributions and generalstatistics and often find representations in terms of powerseriesThe quantile function for a probability distribution hasmany uses in both the theory and application of probabilityIt may be used to generate values of a random variable having119865(119909) as its distributions functionThis fact serves as the basisof a method for simulating a sample from an arbitrary distri-bution with the aid of a uniform random number generator
The quantile function of 119883 say 119909 = 119876BGHN(119906) = 119865minus1
(119906)can be obtained by inverting the cumulative function (3)Now we provide a power series expansion for 119876BGHN(119906)that can be useful to determine some mathematical mea-sures of the BGHN distribution First an expansion forthe inverse of the incomplete beta function 119868
119909(119886 119887) = 119906
can be found in wolfram website (httpfunctionswolframcom062306000401) as
119911 = 119876119861(119906) =
infin
sum
119894=0
119889119894119906119894119886
(12)
where119876119861(119906) is the beta quantile function 119889
119894= 119890
119894[119886119861(119886 119887)]
1119886
for 119886 gt 0 and 1198900= 0 119890
1= 1 119890
2= (119887 minus 1)(119886 + 1)
Journal of Probability and Statistics 3
The coefficients 1198901015840119894s for 119894 ge 2 can be obtained from a cubic
recursion of the form
119890119894=
1
[1198942+ (120572 minus 2) 119894 + (1 minus 120572)]
times (1 minus 1205751198942)
119894minus1
sum
119903=2
119890119903119890119894+1minus119903
times [119903 (1 minus 119886) (119894 minus 119903) minus 119903 (119903 minus 1)]
+
119894minus1
sum
119903=1
119894minus119903
sum
119904=1
119890119903119890119904119890119894+1minus119903minus119904
times [119903 (119903 minus 119886) + 119904 (119886 + 119887 minus 2) (119894 + 1 minus 119903 minus 119904)]
(13)
where 1205751198942= 1 if 119894 = 2 and 120575
1198942= 0 if 119894 = 2 In the last equation
we note that the quadratic term only contributes for 119894 ge 3Following Steinbrecher [4] the quantile function of the
standard normal distribution say 119876119873(119906) = Φ
minus1
(119906) can beexpanded as
119876119873(119906) = Φ
minus1
(119906) =
infin
sum
119896=0
1198871198961205772119896+1
(14)
where 120577 = radic2120587(119906 minus 12) and the 1198871015840119896s can be calculated from
1198870= 1 and
119887119896+1
=
1
2 (2119896 + 3)
119896
sum
119903=0
(2119903 + 1) (2119896 minus 2119903 + 1) 119887119903119887119896minus119903
(119903 + 1) (2119903 + 1)
(15)
Here 1198871= 16 119887
2= 7120 119887
3= 1277560
The function 119876119873(119906) can be expressed as a power series
given by
119876119873(119906) =
infin
sum
119895=0
119891119895119906119895
(16)
where 119891119895= sum
infin
119896=119895(minus12)
119896minus119895
(119896
119895) 119892
119896and the quantities 119892
119896are
defined from the coefficients in (14) by 119892119896
= 0 for 119896 =
0 2 4 and 119892119896= (2120587)
1198962
119887(119896minus1)2
for 119896 = 1 3 5 By inverting the normal cumulative function in 119911 =
2Φ[(119909120579)120572
] minus 1 and using (16) we can express the quantilefunction of119883 in terms of 119911 as
119909 = 120579[119876119873(
119911 + 1
2
)]
1120572
= 120579[
[
infin
sum
119895=0
119891119895(
119911 + 1
2
)
119895
]
]
1120572
= 120579[
[
infin
sum
119895=0
119895
sum
119901=0
119891119895
2119895(
119895
119901) 119911
119901]
]
1120572
(17)
Replacingsuminfin
119895=0sum119895
119901=0bysuminfin
119901=0suminfin
119895=119901in last equation we obtain
119909 = 120579 (
infin
sum
119901=0
ℎ119901119911119901
)
1120572
(18)
where ℎ119901= sum
infin
119895=1199012minus119895
119891119895(119895
119901) Using the same steps of (8) we
can write
(
infin
sum
119901=0
ℎ119901119911119901
)
1120572
=
infin
sum
119903=0
119904119903(120572
minus1
)(
infin
sum
119901=0
ℎ119901119911119901
)
119903
(19)
where 119904119903(120572
minus1
) = suminfin
119895=119903(minus1)
119903+119895
(120572minus1
119895) (
119895
119903)
We use throughout an equation of Gradshteyn andRyzhik [5] for a power series raised to a positive integer 119895
(
infin
sum
119894=0
119886119894119909119894
)
119895
=
infin
sum
119894=0
119888119895119894119909119894
(20)
where the coefficients 119888119895119894(for 119894 = 1 2 ) are easily obtained
from the recurrence equation
119888119895119894= (119894119886
0)minus1
119894
sum
119898=1
[119898 (119895 + 1) minus 119894] 119886119898119888119895119894minus119898
(21)
and 1198881198950
= 119886119895
0 From (19) and (20) we have
119909 =
infin
sum
119901=0
ℎ⋆
119901119911119901
(22)
where ℎ⋆
119901= 120579sum
infin
119903=0119904119903(120572
minus1
) ℎ119903119901
for 119901 ge 0 ℎ119903119901
=
(119901ℎ0)minus1
sum119901
119898=1[119898(119903 + 1) minus 119901] ℎ
119898ℎ119903119901minus119898
for 119901 ge 1 and ℎ1199030
= ℎ119903
0
By inserting (12) in (22) we obtain
119909 = 119876BGHN (119906) =infin
sum
119901=0
ℎ⋆
119901(
infin
sum
119894=0
119889119894119906119894119886
)
119901
(23)
From (20) it follows (suminfin
119894=0119889119894119906119894119886
)
119901
= suminfin
119894=0119889119901119894119906119894119886 where the
quantities 119889119901119894
are determined recursively from 1198890119894= 119889
119894
0and
119889119901119894
= (1198941198890)minus1
sum119894
119898=1[119898 (119901 + 1) minus 119894]119889
119898119889119901119894minus119898
(for 119894 = 1 2 )Finally we obtain
119909 = 119876BGHN (119906) =infin
sum
119894=0
V119894119906119894119886
(24)
where V119894= sum
infin
119901=0ℎ⋆
119901119889119901119894 Equation (24) gives the BGHN
quantile function as a power series and represents the mainresult of this section
3 BGHN Properties
31Moments andGenerating Function Here we provide newexpressions for the moments and mgf of 119883 based upon thepower series for its quantile function as alternative resultsform those obtained by Pescim et al [3]
We can write from (24)
1205831015840
119904(120572 120579 119886 119887) = 119864 (119883
119904
) = int
infin
0
119909119904
119891 (119909) 119889119909
= int
1
0
(
infin
sum
119894=0
V119894119906119894119886
)
119904
119889119906
(25)
4 Journal of Probability and Statistics
and then using (20) we obtain
1205831015840
119904(120572 120579 119886 119887) = 119886
infin
sum
119894=0
V119904119894
(119894 + 119886)
(26)
where the quantities V119904119894(for 119894 = 1 2 ) are easily determined
from the recurrence equation
V119904119894= (119894V
0)minus1
119894
sum
119898=1
[119898 (119904 + 1) minus 119894] V119898V119904119894minus119898
(27)
with V1199040
= V1199040
We now provide a new alternative representation for themgf of 119883 say 119872
120572120579119886119887(119905) = 119864(119890
119905119883
) based on the quantilepower series (24) We can write
119872120572120579119886119887
(119905) = int
infin
0
119890119905119909
119891 (119909) 119889119909
= int
1
0
exp[119905(infin
sum
119894=0
V119894119906119894119886
)]119889119906
(28)
We expand the exponential function and use the same algebrathat leads to (26)
119872120572120579119886119887
(119905) =
infin
sum
119903=0
119905119903
119903
int
1
0
(
infin
sum
119894=0
V119894119906119894119886
)
119903
119889119906
=
infin
sum
119903119894=0
119905119903V119903119894
119903
int
1
0
119906119894119886
119889119906
(29)
and then
119872120572120579119886119887
(119905) = 119886
infin
sum
119903119894=0
119905119903V119903119894
(119886 + 119894) 119903
(30)
Equations (26) and (30) are the main results of this section
32 Mean andMedian Deviations The amount of scatter in apopulation is evidently measured to some extent by the meandeviations in relation to the mean and the median defined by
1205751(119883) = int
infin
0
10038161003816100381610038161003816119909 minus 120583
1015840
1
10038161003816100381610038161003816119891 (119909) 119889119909
1205752(119883) = int
infin
0
|119909 minus119872|119891 (119909) 119889119909
(31)
respectively where 1205831015840
1= 119864(119883) and 119872 = Median(119883)
denotes the median Here119872 is calculated as the solution ofthe nonlinear equation 119868
2Φ[(119872120579)120572]minus1(119886 119887) = 12 We define
119879(119902) = int
infin
119902
119909119891(119909)119889119909 which is determined below The mea-sures 120575
1(119883) and 120575
2(119883) can be written in terms of 1205831015840
1and 119879(119902)
as
1205751(119883) = 2120583
1015840
1119865 (120583
1015840
1) minus 2120583
1015840
1+ 2119879 (120583
1015840
1)
1205752(119883) = 2119879 (119872) minus 120583
1015840
1
(32)
For more details see Paranaıba et al [6] Clearly 119865(119872) and119865(120583
1015840
1) are determined from (3) From (10) we have
119879 (119902)
= 120572radic2
120587
infin
sum
119903=0
119887119903int
infin
119902
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
erf [(119909120579)120572
radic2
]
119903
119889119909
(33)
Setting 119906 = (119909120579)120572 in the last equation gives
119879 (119902)
= 120579radic2
120587
infin
sum
119903=0
119887119903int
infin
(119902120579)1205721199061120572
119890minus11990622
[erf ( 119906
radic2
)]
119903
119889119906
(34)
Using the power series for the error function erf(119909) =
(2radic120587)suminfin
119898=0((minus1)
119898
1199092119898+1
(2119898 + 1)119898) (see eg [7]) weobtain after some algebra
119879 (119902) = 120579radic2
120587
infin
sum
119903=0
119887119903(
2
radic120587
)
119903
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119903=0
(minus1)1198981+sdotsdotsdot+119898
119903
(21198981+ 1) sdot sdot sdot (2119898
119903+ 1)119898
1 sdot sdot sdot 119898
119903
times Γ [(1198981+ sdot sdot sdot + 119898
119903+
119903
2
+
1
2120572
+
1
2
)
1
2
(
119902
120579
)
120572
]
(35)
where Γ(119901 119909) = int
infin
119909
V119901minus1119890minusV119889V denotes the complementaryincomplete gamma function for 119901 gt 0 The measures 120575
1(119883)
and 1205752(119883) are immediately calculated from (35)
Bonferroni and Lorenz curves have applications not onlyin economics to study income and poverty but also inother fields such as reliability demography insurance andmedicine They are defined by
119861 (120588) =
1
1205881205831015840
1
int
119902
0
119909119891 (119909) 119889119909
119871 (120588) =
1
1205831015840
1
int
119902
0
119909119891 (119909) 119889119909
(36)
respectively where 119902 = 119876BGHN(120588) = 119876119861(]) and ] =
2Φ[(120588120579)120572
] minus 1 (Section 2) for a given probability 120588 Fromint
119902
0
119909119891(119909)119889119909 = 1205831015840
1minus 119879(120588) we obtain 119861(120588) = 120588
minus1
[1 minus 119879(120588)1205831015840
1]
and 119871(120588) = 1 minus 119879(120588)1205831015840
1
33 Renyi Entropy The Renyi information of order 120585 for acontinuous random variable with density function 119891(119909) isdefined as
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (37)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 Applicationsof the Renyi entropy can be found in several areas suchas physics information theory and engineering to describe
Journal of Probability and Statistics 5
many nonlinear dynamical or chaotic systems [8] and instatistics as certain appropriately scaled test statistics (relativeRenyi information) for testing hypotheses in parametricmodels [9] Renyi [10] generalized the concept of informationtheory which allows for different averaging of probabilitiesvia 120585
For the BGHN distribution (4) the Renyi entropy isdefined by
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (38)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 From (4) we have
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
120585(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120585(119887minus1)
119889119909
(39)
For |119911| lt 1 and 119887 is a real noninteger the power series holds
(1 minus 119911)119887minus1
=
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119911
119895
(40)
where the binomial coefficient is defined for any real Using(40) in (39) twice 119868(120585) can be expressed as
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times
infin
sum
119895119896=0
(minus1)119895+119896
2119895
(
120585 (119886 minus 1) + 1
119895)
times (
120585 (119887 minus 1) + 119895 + 1
119896)
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
Φ[(
119909
120579
)
120572
]
119896
119889119909
(41)
Substituting Φ(119909) by the error function and setting 119906 =
(119909120579)120572 119868(120585) reduces to
119868 (120585) = 120572120585minus1
1205791minus120585
2120585(119887minus1)
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times int
infin
0
119906((120585(120572minus1)+1)120572)minus1
119890minus(1205852)119906
2
[erf ( 119906
radic2
)]
119897
119889119906
(42)
Following similar algebra that lead to (35) we obtain
119868 (120585) = 120572120585minus1
1205791minus120585
2[120585(119887minus1)+(120585(120572minus1)+1)2120572minus1]
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
(21198981+ 1) sdot sdot sdot (2119898
119897+ 1)119898
1 sdot sdot sdot 119898
119897
times 120585minus[1198981+sdotsdotsdot+119898
119897+1198972+(120585(120572minus1)+1)2120572]
times Γ(1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
)
(43)
Finally the Renyi entropy reduces to
120485119877(120585) = (1 minus 120585)
minus1
times
(120585 minus 1) log (120572) + (1 minus 120585) log (120579)
+ [120585 (119887 minus 1) +
120585 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120585 log(radic 2
120587
) + log[
[
infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)]
]
+ log[infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
times ((21198981+ 1) sdot sdot sdot (2119898
119897+ 1)
times 1198981 sdot sdot sdot 119898
119897)minus1
]
minus [1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
] log (120585)
+ log [Γ(1198981+sdot sdot sdot+119898
119897+
119897
2
+
120585 (120572minus1) + 1
2120572
)]
(44)
34 Reliability In the context of reliability the stress-strengthmodel describes the life of a component which has a randomstrength 119883
1that is subjected to a random stress 119883
2 The
component fails at the instant that the stress applied toit exceeds the strength and the component will functionsatisfactorily whenever 119883
1gt 119883
2 Hence 119877 = Pr(119883
2lt
1198831) is a measure of component reliability Here we derive
119877 when 1198831and 119883
2have independent BGHN(120572 120579 119886
1 1198871)
and BGHN(120572 120579 1198862 1198872) distributions with the same shape
parameters 120572 and 120579 The reliability 119877 becomes
119877 = int
infin
0
1198911(119909) 119865
2(119909) 119889119909 (45)
6 Journal of Probability and Statistics
where the cdf of 1198832and the density of 119883
1are obtained from
(6) and (10) as
1198652(119909) =
infin
sum
119895=0
119908119895(119886
2 1198872) 2Φ [(
119909
120579
)
120572
] minus 1
1198862+119895
1198911(119909) = radic
2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times
infin
sum
119903=0
119887119903(119886
1 1198871) 2Φ [(
119909
120579
)
120572
] minus 1
119903
(46)
respectively where
119908119895(119886
2 1198872) =
(minus1)119895
119861 (1198862 1198872)
(
1198872minus 1
119895)
119887119903(119886
1 1198871) =
infin
sum
119895=0
(minus1)119895
119861 (1198861 1198871)
(
1198871minus 1
119895) 119904
119903(119886
1+ 119895 minus 1)
(47)
refer to1198832and119883
1 respectively Hence
119877 = 120572radic2
120587
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
times int
infin
0
119909minus1
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
1198862+119895+119903
119889119909
(48)
Setting 119906 = 2Φ[(119909120579)120572
] minus 1 in the last integral the reli-ability of119883 reduces to
119877 =
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
1198862+ 119895 + 119903 + 1
(49)
4 Computational Issues
Here we show the practical use of (24) (26) (30) (35) and(49) If we truncate these infinite power series we have asimple way to compute the quantile function moments mgfmean deviations and reliability of the BGHN distributionThe question is how large the truncation limit should be
We now provide evidence that each infinite summationcan be truncated at twenty to yield sufficient accuracy Let119863
1
denote the absolute difference between the integrated version
1
119861 (119886 119887)
int
2Φ[(119909120579)120572]minus1
0
119905119886minus1
(1 minus 119905)119887minus1
119889119905 = 119880 (50)
where 119880 sim 119880(0 1) is a uniform variate on the unit intervaland the truncated version of (24) averaged over (all with step001) 119909 = 001 5 119886 = 001 10 119887 = 001 10
120572 = 001 10 and 120579 = 001 10 Let 1198632denote the
absolute difference between integrated version
1205831015840
119904=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
119909119904
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(51)
and the truncated version of (26) averaged over 119909 =
001 5 119886 = 001 10 119887 = 001 10120572 = 001 10
and 120579 = 001 10 Let 1198633denote the absolute difference
between the truncated version of (30) and the integratedversion
119872120572120579119886119887
(119905)
=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
(
120572
119909
)(
119909
120579
)
120572
exp [119905119909 minus 1
2
(
119909
120579
)
2120572
]
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(52)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198634denote
the absolute difference between the truncated version of (35)and the integrated version
119879 (119902) =
120572 2119887minus1
119861 (119886 119887)
radic2
120587
int
119902
0
(
119909
120579
)
120572
times exp [minus12
(
119909
120579
)
2120572
] 2Φ[(
119909
120579
)
120572
]minus1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(53)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198635denote
the absolute difference between the truncated version of (49)and the integrated version (45) averaged over 119909 = 001 5119886 = 001 10 119887 = 001 10 120572 = 001 10 and 120579 =
001 10We obtain the following estimates after extensive compu-
tations1198631= 231times10
minus201198632= 847times10
minus181198633= 122times10
minus211198634= 151 times 10
minus22 and1198635= 941 times 10
minus19 These estimates aresmall enough to suggest that the truncated versions of (24)(26) (30) (35) and (49) are reasonable practical use
It would be important to verify that each (untruncated)infinite series (such as (24) (26) (30) (35) and (49)) is con-vergent and provide valid values for its arguments Howeverthis will be a difficult mathematical problem that could beinvestigated in future work
Journal of Probability and Statistics 7
5 Properties of the BGHN Order Statistics
Order statistics have been used in a wide range of prob-lems including robust statistical estimation and detectionof outliers characterization of probability distributions andgoodness-of-fit tests entropy estimation analysis of censoredsamples reliability analysis quality control and strength ofmaterials
51 Mixture Form Suppose 1198831 119883
119899is a random sample
of size 119899 from a continuous distribution and let1198831119899
lt 1198832119899
lt
sdot sdot sdot lt 119883119899119899
denote the corresponding order statistics Therehas been a large amount of work relating tomoments of orderstatistics119883
119894119899 See Arnold et al [11] David and Nagaraja [12]
and Ahsanullah and Nevzorov [13] for excellent accounts Itis well known that the density function of119883
119894119899is given by
119891119894119899(119909) =
119891 (119909)
119861 (119894 119899 minus 119894 + 1)
119865(119909)119894minus1
[1 minus 119865 (119909)]119899minus119894
(54)
For the BGHN distribution Pescim et al [3] obtained
119891119894119899(119909) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119891119896119872119896(119909) (55)
where 119872119896denotes a sequence (119898
1 119898
119894+119896minus1) of 119894 + 119896 minus 1
nonnegative integers 119891119896119872119896
(119909) denotes the BGHN(120572 120579 (119894 +119896)119886 + sum
119894+119896minus1
119895=1119898119895 119887) density function defined under 119872
119896and
the constants 120578119896119872119896
are given by
120578119896119872119896
=
(minus1)119896+sum119894+119896minus1
119895=1119898119895(119899minus119894
119896) 119861 (119886 (119894 + 119896)+sum
119894+119896minus1
119895=1119898119895 119887) Γ(119887)
119894+119896minus1
119861(119886 119887)119894+119896
119861 (119894 119899minus 119894+1)prod119894+119896minus1
119895=1Γ (119887minus119898
119895)119898
119895 (119886+119898
119895)
(56)
The quantities 120578119896119872119896
are easily obtained given 119896 and asequence 119872
119896of indices 119898
1 119898
119894+119896minus1 The sums in (55)
extend over all (119894 + 119896)-tuples (1198961198981 119898
119894+119896minus1) and can be
implementable on a computer If 119887 gt 0 is an integer (55)holds but the indices 119898
1 119898
119894+119896minus1vary from zero to 119887 minus 1
Equation (55) reveals that the density function of the BGHNorder statistics can be expressed as an infinite mixture ofBGHN densities
52 Moments of Order Statistics Themoments of the BGHNorder statistics can be written directly in terms of themoments of BGHNdistributions from themixture form (55)We have
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119864 (119883119904
119894119896) (57)
where119883119894119896sim BGHN(120572 120579 (119894 + 119896)119886 + sum119894+119896minus1
119895=1119898119895 119887) and 119864(119883119904
119894119896)
can be determined from (26) Inserting (26) in (57) andchanging indices we can write
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
infin
sum
119902=0
119886⋆V
119904119902
(119902 + 119886⋆)
(58)
where
119886⋆
= (119894 + 119896) 119886 +
119894+119896minus1
sum
119895=1
119898119895 (59)
The moments 119864(119883119904
119894119899) can be determined based on the
explicit sum (58) using the software Mathematica They arealso computed from (55) by numerical integration using thestatistical software R [14] The values from both techniquesare usually close wheninfin is replaced by a moderate numbersuch as 20 in (58) For some values 119886 = 2 119887 = 3 and 119899 = 5Table 1 lists the numerical values for the moments 119864(119883119904
119894119899)
(119904 = 1 4 119894 = 1 5) and for the variance skewness andkurtosis
53 Generating Function The mgf of the BGHN orderstatistics can be written directly in terms of the BGHNmgf rsquosfrom the mixture form (55) and (30) We obtain
119872119894119899(119905) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119872120572120579119886⋆119887(119905) (60)
where119872120572120579119886⋆119887(119905) is the mgf of the BGHN(120572 120579 119886⋆ 119887) distri-
bution obtained from (30)
54 Mean Deviations The mean deviations of the orderstatistics in relation to the mean and the median are definedby
1205751(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 120583
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
1205752(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 119872
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
(61)
respectively where 120583119894= 119864(119883
119894119899) and 119872
119894= Median(119883
119894119899)
denotes the median Here 119872119894is obtained as the solution of
the nonlinear equation119899
sum
119903=119894
(
119899
119903) 119868
2Φ[(119872119894120579)120572]minus1
(119886 119887)
119903
times 1 minus 1198682Φ[(119872
119894120579)120572]minus1
(119886 119887)
119899minus119903
=
1
2
(62)
The measures 1205751(119883
119894119899) and 120575
2(119883
119894119899) follow from
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(63)
where 119869119894(119902) = int
infin
119902
119909119891119894119899(119909)119889119909 Using (55) we have
119869119894(119902) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119894119872119896
119879 (119902) (64)
where 119879(119902) is obtained from (35) using 119886⋆ given by (59) and
119887⋆
119903= 119887
119903(119886
⋆
119887) =
1
119861 (119886⋆ 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886
⋆
+ 119895 minus 1)
(65)
8 Journal of Probability and Statistics
Table 1 Moments of the BGHN order statistics for 120579 = 85 120572 = 3 119886 = 2 and 119887 = 3
Order statistics rarr 1198831 5
1198832 5
1198833 5
1198834 5
1198835 5
119904 = 1 267942 581946 473975 171571 023289119904 = 2 1810007 3931172 3201807 1159006 157328119904 = 3 1278683 2777184 2261923 8187821 1111451119904 = 4 938314 2037933 1659828 6008327 8155966Variance 1092078 544561 955281 864636 151904Skewness 057767 minus113596 minus054601 127135 536291Kurtosis 161751 406406 189737 291095 3107642
where
119904119903(119886
⋆
+ 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886⋆+ 119895
(
119886⋆
+ 119895 minus 1
119896)(
119896
119903) (66)
Bonferroni and Lorenz curves of the order statistics aregiven by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(67)
respectively where 119902 = 119865minus1
119894119899(120588) for a given probability 120588 From
int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus 119869
119894(120588) we obtain
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(68)
55 Renyi Entropy The Renyi entropy of the order statisticsis defined by
120485119877(120582) =
1
1 minus 120582
log [119867 (120582)] (69)
where 119867(120582) = int119891120582
119894119899(119909)119889119909 120582 gt 0 and 120582 = 1 From (54) it
follows that
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
int
infin
0
119891120582
(119909)
times [119865 (119909)]120582(119894minus1)
[1minus119865 (119909)]120582(119899minus119894)
119889119909
(70)
Using (40) in (70) we obtain
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
infin
sum
1198961=0
(minus1)1198961(
120582 (119894 minus 1)
1198961
)
times int
infin
0
119891120582
(119909) [119865 (119909)]120582(119894minus1)+119896
1119889119909
(71)
For 120582 gt 0 and 120582 = 1 we can expand [119865(119909)]120582(119894minus1)+1198961 as
119865(119909)120582(119894minus1)+119896
1= 1 minus [1 minus 119865 (119909)]
120582(119894minus1)+1198961
=
infin
sum
1199011=0
(minus1)1199011(
120582 (119894 minus 1) + 1198961+ 1
1199011
) [1 minus 119865 (119909)]1199011
(72)
and then
119865(119909)120582(119894minus1)+119896
1=
infin
sum
1199011=0
1199011
sum
1198971=0
(minus1)1199011+1198971
times (
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)119865(119909)1198971
(73)
Hence from (70) we can write
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
(minus1)1198961+1199011+1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
times int
infin
0
119891120582
(119909) 119865(119909)1198971119889119909
(74)
By replacing 119891(119909) and 119865(119909) by (10) and (16) given by Pescimet al [3] we obtain
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
[Γ (119887)]1198971(minus1)
1198961+1199011+1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 3
The coefficients 1198901015840119894s for 119894 ge 2 can be obtained from a cubic
recursion of the form
119890119894=
1
[1198942+ (120572 minus 2) 119894 + (1 minus 120572)]
times (1 minus 1205751198942)
119894minus1
sum
119903=2
119890119903119890119894+1minus119903
times [119903 (1 minus 119886) (119894 minus 119903) minus 119903 (119903 minus 1)]
+
119894minus1
sum
119903=1
119894minus119903
sum
119904=1
119890119903119890119904119890119894+1minus119903minus119904
times [119903 (119903 minus 119886) + 119904 (119886 + 119887 minus 2) (119894 + 1 minus 119903 minus 119904)]
(13)
where 1205751198942= 1 if 119894 = 2 and 120575
1198942= 0 if 119894 = 2 In the last equation
we note that the quadratic term only contributes for 119894 ge 3Following Steinbrecher [4] the quantile function of the
standard normal distribution say 119876119873(119906) = Φ
minus1
(119906) can beexpanded as
119876119873(119906) = Φ
minus1
(119906) =
infin
sum
119896=0
1198871198961205772119896+1
(14)
where 120577 = radic2120587(119906 minus 12) and the 1198871015840119896s can be calculated from
1198870= 1 and
119887119896+1
=
1
2 (2119896 + 3)
119896
sum
119903=0
(2119903 + 1) (2119896 minus 2119903 + 1) 119887119903119887119896minus119903
(119903 + 1) (2119903 + 1)
(15)
Here 1198871= 16 119887
2= 7120 119887
3= 1277560
The function 119876119873(119906) can be expressed as a power series
given by
119876119873(119906) =
infin
sum
119895=0
119891119895119906119895
(16)
where 119891119895= sum
infin
119896=119895(minus12)
119896minus119895
(119896
119895) 119892
119896and the quantities 119892
119896are
defined from the coefficients in (14) by 119892119896
= 0 for 119896 =
0 2 4 and 119892119896= (2120587)
1198962
119887(119896minus1)2
for 119896 = 1 3 5 By inverting the normal cumulative function in 119911 =
2Φ[(119909120579)120572
] minus 1 and using (16) we can express the quantilefunction of119883 in terms of 119911 as
119909 = 120579[119876119873(
119911 + 1
2
)]
1120572
= 120579[
[
infin
sum
119895=0
119891119895(
119911 + 1
2
)
119895
]
]
1120572
= 120579[
[
infin
sum
119895=0
119895
sum
119901=0
119891119895
2119895(
119895
119901) 119911
119901]
]
1120572
(17)
Replacingsuminfin
119895=0sum119895
119901=0bysuminfin
119901=0suminfin
119895=119901in last equation we obtain
119909 = 120579 (
infin
sum
119901=0
ℎ119901119911119901
)
1120572
(18)
where ℎ119901= sum
infin
119895=1199012minus119895
119891119895(119895
119901) Using the same steps of (8) we
can write
(
infin
sum
119901=0
ℎ119901119911119901
)
1120572
=
infin
sum
119903=0
119904119903(120572
minus1
)(
infin
sum
119901=0
ℎ119901119911119901
)
119903
(19)
where 119904119903(120572
minus1
) = suminfin
119895=119903(minus1)
119903+119895
(120572minus1
119895) (
119895
119903)
We use throughout an equation of Gradshteyn andRyzhik [5] for a power series raised to a positive integer 119895
(
infin
sum
119894=0
119886119894119909119894
)
119895
=
infin
sum
119894=0
119888119895119894119909119894
(20)
where the coefficients 119888119895119894(for 119894 = 1 2 ) are easily obtained
from the recurrence equation
119888119895119894= (119894119886
0)minus1
119894
sum
119898=1
[119898 (119895 + 1) minus 119894] 119886119898119888119895119894minus119898
(21)
and 1198881198950
= 119886119895
0 From (19) and (20) we have
119909 =
infin
sum
119901=0
ℎ⋆
119901119911119901
(22)
where ℎ⋆
119901= 120579sum
infin
119903=0119904119903(120572
minus1
) ℎ119903119901
for 119901 ge 0 ℎ119903119901
=
(119901ℎ0)minus1
sum119901
119898=1[119898(119903 + 1) minus 119901] ℎ
119898ℎ119903119901minus119898
for 119901 ge 1 and ℎ1199030
= ℎ119903
0
By inserting (12) in (22) we obtain
119909 = 119876BGHN (119906) =infin
sum
119901=0
ℎ⋆
119901(
infin
sum
119894=0
119889119894119906119894119886
)
119901
(23)
From (20) it follows (suminfin
119894=0119889119894119906119894119886
)
119901
= suminfin
119894=0119889119901119894119906119894119886 where the
quantities 119889119901119894
are determined recursively from 1198890119894= 119889
119894
0and
119889119901119894
= (1198941198890)minus1
sum119894
119898=1[119898 (119901 + 1) minus 119894]119889
119898119889119901119894minus119898
(for 119894 = 1 2 )Finally we obtain
119909 = 119876BGHN (119906) =infin
sum
119894=0
V119894119906119894119886
(24)
where V119894= sum
infin
119901=0ℎ⋆
119901119889119901119894 Equation (24) gives the BGHN
quantile function as a power series and represents the mainresult of this section
3 BGHN Properties
31Moments andGenerating Function Here we provide newexpressions for the moments and mgf of 119883 based upon thepower series for its quantile function as alternative resultsform those obtained by Pescim et al [3]
We can write from (24)
1205831015840
119904(120572 120579 119886 119887) = 119864 (119883
119904
) = int
infin
0
119909119904
119891 (119909) 119889119909
= int
1
0
(
infin
sum
119894=0
V119894119906119894119886
)
119904
119889119906
(25)
4 Journal of Probability and Statistics
and then using (20) we obtain
1205831015840
119904(120572 120579 119886 119887) = 119886
infin
sum
119894=0
V119904119894
(119894 + 119886)
(26)
where the quantities V119904119894(for 119894 = 1 2 ) are easily determined
from the recurrence equation
V119904119894= (119894V
0)minus1
119894
sum
119898=1
[119898 (119904 + 1) minus 119894] V119898V119904119894minus119898
(27)
with V1199040
= V1199040
We now provide a new alternative representation for themgf of 119883 say 119872
120572120579119886119887(119905) = 119864(119890
119905119883
) based on the quantilepower series (24) We can write
119872120572120579119886119887
(119905) = int
infin
0
119890119905119909
119891 (119909) 119889119909
= int
1
0
exp[119905(infin
sum
119894=0
V119894119906119894119886
)]119889119906
(28)
We expand the exponential function and use the same algebrathat leads to (26)
119872120572120579119886119887
(119905) =
infin
sum
119903=0
119905119903
119903
int
1
0
(
infin
sum
119894=0
V119894119906119894119886
)
119903
119889119906
=
infin
sum
119903119894=0
119905119903V119903119894
119903
int
1
0
119906119894119886
119889119906
(29)
and then
119872120572120579119886119887
(119905) = 119886
infin
sum
119903119894=0
119905119903V119903119894
(119886 + 119894) 119903
(30)
Equations (26) and (30) are the main results of this section
32 Mean andMedian Deviations The amount of scatter in apopulation is evidently measured to some extent by the meandeviations in relation to the mean and the median defined by
1205751(119883) = int
infin
0
10038161003816100381610038161003816119909 minus 120583
1015840
1
10038161003816100381610038161003816119891 (119909) 119889119909
1205752(119883) = int
infin
0
|119909 minus119872|119891 (119909) 119889119909
(31)
respectively where 1205831015840
1= 119864(119883) and 119872 = Median(119883)
denotes the median Here119872 is calculated as the solution ofthe nonlinear equation 119868
2Φ[(119872120579)120572]minus1(119886 119887) = 12 We define
119879(119902) = int
infin
119902
119909119891(119909)119889119909 which is determined below The mea-sures 120575
1(119883) and 120575
2(119883) can be written in terms of 1205831015840
1and 119879(119902)
as
1205751(119883) = 2120583
1015840
1119865 (120583
1015840
1) minus 2120583
1015840
1+ 2119879 (120583
1015840
1)
1205752(119883) = 2119879 (119872) minus 120583
1015840
1
(32)
For more details see Paranaıba et al [6] Clearly 119865(119872) and119865(120583
1015840
1) are determined from (3) From (10) we have
119879 (119902)
= 120572radic2
120587
infin
sum
119903=0
119887119903int
infin
119902
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
erf [(119909120579)120572
radic2
]
119903
119889119909
(33)
Setting 119906 = (119909120579)120572 in the last equation gives
119879 (119902)
= 120579radic2
120587
infin
sum
119903=0
119887119903int
infin
(119902120579)1205721199061120572
119890minus11990622
[erf ( 119906
radic2
)]
119903
119889119906
(34)
Using the power series for the error function erf(119909) =
(2radic120587)suminfin
119898=0((minus1)
119898
1199092119898+1
(2119898 + 1)119898) (see eg [7]) weobtain after some algebra
119879 (119902) = 120579radic2
120587
infin
sum
119903=0
119887119903(
2
radic120587
)
119903
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119903=0
(minus1)1198981+sdotsdotsdot+119898
119903
(21198981+ 1) sdot sdot sdot (2119898
119903+ 1)119898
1 sdot sdot sdot 119898
119903
times Γ [(1198981+ sdot sdot sdot + 119898
119903+
119903
2
+
1
2120572
+
1
2
)
1
2
(
119902
120579
)
120572
]
(35)
where Γ(119901 119909) = int
infin
119909
V119901minus1119890minusV119889V denotes the complementaryincomplete gamma function for 119901 gt 0 The measures 120575
1(119883)
and 1205752(119883) are immediately calculated from (35)
Bonferroni and Lorenz curves have applications not onlyin economics to study income and poverty but also inother fields such as reliability demography insurance andmedicine They are defined by
119861 (120588) =
1
1205881205831015840
1
int
119902
0
119909119891 (119909) 119889119909
119871 (120588) =
1
1205831015840
1
int
119902
0
119909119891 (119909) 119889119909
(36)
respectively where 119902 = 119876BGHN(120588) = 119876119861(]) and ] =
2Φ[(120588120579)120572
] minus 1 (Section 2) for a given probability 120588 Fromint
119902
0
119909119891(119909)119889119909 = 1205831015840
1minus 119879(120588) we obtain 119861(120588) = 120588
minus1
[1 minus 119879(120588)1205831015840
1]
and 119871(120588) = 1 minus 119879(120588)1205831015840
1
33 Renyi Entropy The Renyi information of order 120585 for acontinuous random variable with density function 119891(119909) isdefined as
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (37)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 Applicationsof the Renyi entropy can be found in several areas suchas physics information theory and engineering to describe
Journal of Probability and Statistics 5
many nonlinear dynamical or chaotic systems [8] and instatistics as certain appropriately scaled test statistics (relativeRenyi information) for testing hypotheses in parametricmodels [9] Renyi [10] generalized the concept of informationtheory which allows for different averaging of probabilitiesvia 120585
For the BGHN distribution (4) the Renyi entropy isdefined by
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (38)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 From (4) we have
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
120585(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120585(119887minus1)
119889119909
(39)
For |119911| lt 1 and 119887 is a real noninteger the power series holds
(1 minus 119911)119887minus1
=
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119911
119895
(40)
where the binomial coefficient is defined for any real Using(40) in (39) twice 119868(120585) can be expressed as
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times
infin
sum
119895119896=0
(minus1)119895+119896
2119895
(
120585 (119886 minus 1) + 1
119895)
times (
120585 (119887 minus 1) + 119895 + 1
119896)
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
Φ[(
119909
120579
)
120572
]
119896
119889119909
(41)
Substituting Φ(119909) by the error function and setting 119906 =
(119909120579)120572 119868(120585) reduces to
119868 (120585) = 120572120585minus1
1205791minus120585
2120585(119887minus1)
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times int
infin
0
119906((120585(120572minus1)+1)120572)minus1
119890minus(1205852)119906
2
[erf ( 119906
radic2
)]
119897
119889119906
(42)
Following similar algebra that lead to (35) we obtain
119868 (120585) = 120572120585minus1
1205791minus120585
2[120585(119887minus1)+(120585(120572minus1)+1)2120572minus1]
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
(21198981+ 1) sdot sdot sdot (2119898
119897+ 1)119898
1 sdot sdot sdot 119898
119897
times 120585minus[1198981+sdotsdotsdot+119898
119897+1198972+(120585(120572minus1)+1)2120572]
times Γ(1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
)
(43)
Finally the Renyi entropy reduces to
120485119877(120585) = (1 minus 120585)
minus1
times
(120585 minus 1) log (120572) + (1 minus 120585) log (120579)
+ [120585 (119887 minus 1) +
120585 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120585 log(radic 2
120587
) + log[
[
infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)]
]
+ log[infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
times ((21198981+ 1) sdot sdot sdot (2119898
119897+ 1)
times 1198981 sdot sdot sdot 119898
119897)minus1
]
minus [1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
] log (120585)
+ log [Γ(1198981+sdot sdot sdot+119898
119897+
119897
2
+
120585 (120572minus1) + 1
2120572
)]
(44)
34 Reliability In the context of reliability the stress-strengthmodel describes the life of a component which has a randomstrength 119883
1that is subjected to a random stress 119883
2 The
component fails at the instant that the stress applied toit exceeds the strength and the component will functionsatisfactorily whenever 119883
1gt 119883
2 Hence 119877 = Pr(119883
2lt
1198831) is a measure of component reliability Here we derive
119877 when 1198831and 119883
2have independent BGHN(120572 120579 119886
1 1198871)
and BGHN(120572 120579 1198862 1198872) distributions with the same shape
parameters 120572 and 120579 The reliability 119877 becomes
119877 = int
infin
0
1198911(119909) 119865
2(119909) 119889119909 (45)
6 Journal of Probability and Statistics
where the cdf of 1198832and the density of 119883
1are obtained from
(6) and (10) as
1198652(119909) =
infin
sum
119895=0
119908119895(119886
2 1198872) 2Φ [(
119909
120579
)
120572
] minus 1
1198862+119895
1198911(119909) = radic
2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times
infin
sum
119903=0
119887119903(119886
1 1198871) 2Φ [(
119909
120579
)
120572
] minus 1
119903
(46)
respectively where
119908119895(119886
2 1198872) =
(minus1)119895
119861 (1198862 1198872)
(
1198872minus 1
119895)
119887119903(119886
1 1198871) =
infin
sum
119895=0
(minus1)119895
119861 (1198861 1198871)
(
1198871minus 1
119895) 119904
119903(119886
1+ 119895 minus 1)
(47)
refer to1198832and119883
1 respectively Hence
119877 = 120572radic2
120587
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
times int
infin
0
119909minus1
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
1198862+119895+119903
119889119909
(48)
Setting 119906 = 2Φ[(119909120579)120572
] minus 1 in the last integral the reli-ability of119883 reduces to
119877 =
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
1198862+ 119895 + 119903 + 1
(49)
4 Computational Issues
Here we show the practical use of (24) (26) (30) (35) and(49) If we truncate these infinite power series we have asimple way to compute the quantile function moments mgfmean deviations and reliability of the BGHN distributionThe question is how large the truncation limit should be
We now provide evidence that each infinite summationcan be truncated at twenty to yield sufficient accuracy Let119863
1
denote the absolute difference between the integrated version
1
119861 (119886 119887)
int
2Φ[(119909120579)120572]minus1
0
119905119886minus1
(1 minus 119905)119887minus1
119889119905 = 119880 (50)
where 119880 sim 119880(0 1) is a uniform variate on the unit intervaland the truncated version of (24) averaged over (all with step001) 119909 = 001 5 119886 = 001 10 119887 = 001 10
120572 = 001 10 and 120579 = 001 10 Let 1198632denote the
absolute difference between integrated version
1205831015840
119904=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
119909119904
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(51)
and the truncated version of (26) averaged over 119909 =
001 5 119886 = 001 10 119887 = 001 10120572 = 001 10
and 120579 = 001 10 Let 1198633denote the absolute difference
between the truncated version of (30) and the integratedversion
119872120572120579119886119887
(119905)
=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
(
120572
119909
)(
119909
120579
)
120572
exp [119905119909 minus 1
2
(
119909
120579
)
2120572
]
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(52)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198634denote
the absolute difference between the truncated version of (35)and the integrated version
119879 (119902) =
120572 2119887minus1
119861 (119886 119887)
radic2
120587
int
119902
0
(
119909
120579
)
120572
times exp [minus12
(
119909
120579
)
2120572
] 2Φ[(
119909
120579
)
120572
]minus1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(53)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198635denote
the absolute difference between the truncated version of (49)and the integrated version (45) averaged over 119909 = 001 5119886 = 001 10 119887 = 001 10 120572 = 001 10 and 120579 =
001 10We obtain the following estimates after extensive compu-
tations1198631= 231times10
minus201198632= 847times10
minus181198633= 122times10
minus211198634= 151 times 10
minus22 and1198635= 941 times 10
minus19 These estimates aresmall enough to suggest that the truncated versions of (24)(26) (30) (35) and (49) are reasonable practical use
It would be important to verify that each (untruncated)infinite series (such as (24) (26) (30) (35) and (49)) is con-vergent and provide valid values for its arguments Howeverthis will be a difficult mathematical problem that could beinvestigated in future work
Journal of Probability and Statistics 7
5 Properties of the BGHN Order Statistics
Order statistics have been used in a wide range of prob-lems including robust statistical estimation and detectionof outliers characterization of probability distributions andgoodness-of-fit tests entropy estimation analysis of censoredsamples reliability analysis quality control and strength ofmaterials
51 Mixture Form Suppose 1198831 119883
119899is a random sample
of size 119899 from a continuous distribution and let1198831119899
lt 1198832119899
lt
sdot sdot sdot lt 119883119899119899
denote the corresponding order statistics Therehas been a large amount of work relating tomoments of orderstatistics119883
119894119899 See Arnold et al [11] David and Nagaraja [12]
and Ahsanullah and Nevzorov [13] for excellent accounts Itis well known that the density function of119883
119894119899is given by
119891119894119899(119909) =
119891 (119909)
119861 (119894 119899 minus 119894 + 1)
119865(119909)119894minus1
[1 minus 119865 (119909)]119899minus119894
(54)
For the BGHN distribution Pescim et al [3] obtained
119891119894119899(119909) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119891119896119872119896(119909) (55)
where 119872119896denotes a sequence (119898
1 119898
119894+119896minus1) of 119894 + 119896 minus 1
nonnegative integers 119891119896119872119896
(119909) denotes the BGHN(120572 120579 (119894 +119896)119886 + sum
119894+119896minus1
119895=1119898119895 119887) density function defined under 119872
119896and
the constants 120578119896119872119896
are given by
120578119896119872119896
=
(minus1)119896+sum119894+119896minus1
119895=1119898119895(119899minus119894
119896) 119861 (119886 (119894 + 119896)+sum
119894+119896minus1
119895=1119898119895 119887) Γ(119887)
119894+119896minus1
119861(119886 119887)119894+119896
119861 (119894 119899minus 119894+1)prod119894+119896minus1
119895=1Γ (119887minus119898
119895)119898
119895 (119886+119898
119895)
(56)
The quantities 120578119896119872119896
are easily obtained given 119896 and asequence 119872
119896of indices 119898
1 119898
119894+119896minus1 The sums in (55)
extend over all (119894 + 119896)-tuples (1198961198981 119898
119894+119896minus1) and can be
implementable on a computer If 119887 gt 0 is an integer (55)holds but the indices 119898
1 119898
119894+119896minus1vary from zero to 119887 minus 1
Equation (55) reveals that the density function of the BGHNorder statistics can be expressed as an infinite mixture ofBGHN densities
52 Moments of Order Statistics Themoments of the BGHNorder statistics can be written directly in terms of themoments of BGHNdistributions from themixture form (55)We have
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119864 (119883119904
119894119896) (57)
where119883119894119896sim BGHN(120572 120579 (119894 + 119896)119886 + sum119894+119896minus1
119895=1119898119895 119887) and 119864(119883119904
119894119896)
can be determined from (26) Inserting (26) in (57) andchanging indices we can write
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
infin
sum
119902=0
119886⋆V
119904119902
(119902 + 119886⋆)
(58)
where
119886⋆
= (119894 + 119896) 119886 +
119894+119896minus1
sum
119895=1
119898119895 (59)
The moments 119864(119883119904
119894119899) can be determined based on the
explicit sum (58) using the software Mathematica They arealso computed from (55) by numerical integration using thestatistical software R [14] The values from both techniquesare usually close wheninfin is replaced by a moderate numbersuch as 20 in (58) For some values 119886 = 2 119887 = 3 and 119899 = 5Table 1 lists the numerical values for the moments 119864(119883119904
119894119899)
(119904 = 1 4 119894 = 1 5) and for the variance skewness andkurtosis
53 Generating Function The mgf of the BGHN orderstatistics can be written directly in terms of the BGHNmgf rsquosfrom the mixture form (55) and (30) We obtain
119872119894119899(119905) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119872120572120579119886⋆119887(119905) (60)
where119872120572120579119886⋆119887(119905) is the mgf of the BGHN(120572 120579 119886⋆ 119887) distri-
bution obtained from (30)
54 Mean Deviations The mean deviations of the orderstatistics in relation to the mean and the median are definedby
1205751(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 120583
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
1205752(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 119872
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
(61)
respectively where 120583119894= 119864(119883
119894119899) and 119872
119894= Median(119883
119894119899)
denotes the median Here 119872119894is obtained as the solution of
the nonlinear equation119899
sum
119903=119894
(
119899
119903) 119868
2Φ[(119872119894120579)120572]minus1
(119886 119887)
119903
times 1 minus 1198682Φ[(119872
119894120579)120572]minus1
(119886 119887)
119899minus119903
=
1
2
(62)
The measures 1205751(119883
119894119899) and 120575
2(119883
119894119899) follow from
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(63)
where 119869119894(119902) = int
infin
119902
119909119891119894119899(119909)119889119909 Using (55) we have
119869119894(119902) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119894119872119896
119879 (119902) (64)
where 119879(119902) is obtained from (35) using 119886⋆ given by (59) and
119887⋆
119903= 119887
119903(119886
⋆
119887) =
1
119861 (119886⋆ 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886
⋆
+ 119895 minus 1)
(65)
8 Journal of Probability and Statistics
Table 1 Moments of the BGHN order statistics for 120579 = 85 120572 = 3 119886 = 2 and 119887 = 3
Order statistics rarr 1198831 5
1198832 5
1198833 5
1198834 5
1198835 5
119904 = 1 267942 581946 473975 171571 023289119904 = 2 1810007 3931172 3201807 1159006 157328119904 = 3 1278683 2777184 2261923 8187821 1111451119904 = 4 938314 2037933 1659828 6008327 8155966Variance 1092078 544561 955281 864636 151904Skewness 057767 minus113596 minus054601 127135 536291Kurtosis 161751 406406 189737 291095 3107642
where
119904119903(119886
⋆
+ 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886⋆+ 119895
(
119886⋆
+ 119895 minus 1
119896)(
119896
119903) (66)
Bonferroni and Lorenz curves of the order statistics aregiven by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(67)
respectively where 119902 = 119865minus1
119894119899(120588) for a given probability 120588 From
int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus 119869
119894(120588) we obtain
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(68)
55 Renyi Entropy The Renyi entropy of the order statisticsis defined by
120485119877(120582) =
1
1 minus 120582
log [119867 (120582)] (69)
where 119867(120582) = int119891120582
119894119899(119909)119889119909 120582 gt 0 and 120582 = 1 From (54) it
follows that
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
int
infin
0
119891120582
(119909)
times [119865 (119909)]120582(119894minus1)
[1minus119865 (119909)]120582(119899minus119894)
119889119909
(70)
Using (40) in (70) we obtain
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
infin
sum
1198961=0
(minus1)1198961(
120582 (119894 minus 1)
1198961
)
times int
infin
0
119891120582
(119909) [119865 (119909)]120582(119894minus1)+119896
1119889119909
(71)
For 120582 gt 0 and 120582 = 1 we can expand [119865(119909)]120582(119894minus1)+1198961 as
119865(119909)120582(119894minus1)+119896
1= 1 minus [1 minus 119865 (119909)]
120582(119894minus1)+1198961
=
infin
sum
1199011=0
(minus1)1199011(
120582 (119894 minus 1) + 1198961+ 1
1199011
) [1 minus 119865 (119909)]1199011
(72)
and then
119865(119909)120582(119894minus1)+119896
1=
infin
sum
1199011=0
1199011
sum
1198971=0
(minus1)1199011+1198971
times (
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)119865(119909)1198971
(73)
Hence from (70) we can write
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
(minus1)1198961+1199011+1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
times int
infin
0
119891120582
(119909) 119865(119909)1198971119889119909
(74)
By replacing 119891(119909) and 119865(119909) by (10) and (16) given by Pescimet al [3] we obtain
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
[Γ (119887)]1198971(minus1)
1198961+1199011+1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Probability and Statistics
and then using (20) we obtain
1205831015840
119904(120572 120579 119886 119887) = 119886
infin
sum
119894=0
V119904119894
(119894 + 119886)
(26)
where the quantities V119904119894(for 119894 = 1 2 ) are easily determined
from the recurrence equation
V119904119894= (119894V
0)minus1
119894
sum
119898=1
[119898 (119904 + 1) minus 119894] V119898V119904119894minus119898
(27)
with V1199040
= V1199040
We now provide a new alternative representation for themgf of 119883 say 119872
120572120579119886119887(119905) = 119864(119890
119905119883
) based on the quantilepower series (24) We can write
119872120572120579119886119887
(119905) = int
infin
0
119890119905119909
119891 (119909) 119889119909
= int
1
0
exp[119905(infin
sum
119894=0
V119894119906119894119886
)]119889119906
(28)
We expand the exponential function and use the same algebrathat leads to (26)
119872120572120579119886119887
(119905) =
infin
sum
119903=0
119905119903
119903
int
1
0
(
infin
sum
119894=0
V119894119906119894119886
)
119903
119889119906
=
infin
sum
119903119894=0
119905119903V119903119894
119903
int
1
0
119906119894119886
119889119906
(29)
and then
119872120572120579119886119887
(119905) = 119886
infin
sum
119903119894=0
119905119903V119903119894
(119886 + 119894) 119903
(30)
Equations (26) and (30) are the main results of this section
32 Mean andMedian Deviations The amount of scatter in apopulation is evidently measured to some extent by the meandeviations in relation to the mean and the median defined by
1205751(119883) = int
infin
0
10038161003816100381610038161003816119909 minus 120583
1015840
1
10038161003816100381610038161003816119891 (119909) 119889119909
1205752(119883) = int
infin
0
|119909 minus119872|119891 (119909) 119889119909
(31)
respectively where 1205831015840
1= 119864(119883) and 119872 = Median(119883)
denotes the median Here119872 is calculated as the solution ofthe nonlinear equation 119868
2Φ[(119872120579)120572]minus1(119886 119887) = 12 We define
119879(119902) = int
infin
119902
119909119891(119909)119889119909 which is determined below The mea-sures 120575
1(119883) and 120575
2(119883) can be written in terms of 1205831015840
1and 119879(119902)
as
1205751(119883) = 2120583
1015840
1119865 (120583
1015840
1) minus 2120583
1015840
1+ 2119879 (120583
1015840
1)
1205752(119883) = 2119879 (119872) minus 120583
1015840
1
(32)
For more details see Paranaıba et al [6] Clearly 119865(119872) and119865(120583
1015840
1) are determined from (3) From (10) we have
119879 (119902)
= 120572radic2
120587
infin
sum
119903=0
119887119903int
infin
119902
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
erf [(119909120579)120572
radic2
]
119903
119889119909
(33)
Setting 119906 = (119909120579)120572 in the last equation gives
119879 (119902)
= 120579radic2
120587
infin
sum
119903=0
119887119903int
infin
(119902120579)1205721199061120572
119890minus11990622
[erf ( 119906
radic2
)]
119903
119889119906
(34)
Using the power series for the error function erf(119909) =
(2radic120587)suminfin
119898=0((minus1)
119898
1199092119898+1
(2119898 + 1)119898) (see eg [7]) weobtain after some algebra
119879 (119902) = 120579radic2
120587
infin
sum
119903=0
119887119903(
2
radic120587
)
119903
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119903=0
(minus1)1198981+sdotsdotsdot+119898
119903
(21198981+ 1) sdot sdot sdot (2119898
119903+ 1)119898
1 sdot sdot sdot 119898
119903
times Γ [(1198981+ sdot sdot sdot + 119898
119903+
119903
2
+
1
2120572
+
1
2
)
1
2
(
119902
120579
)
120572
]
(35)
where Γ(119901 119909) = int
infin
119909
V119901minus1119890minusV119889V denotes the complementaryincomplete gamma function for 119901 gt 0 The measures 120575
1(119883)
and 1205752(119883) are immediately calculated from (35)
Bonferroni and Lorenz curves have applications not onlyin economics to study income and poverty but also inother fields such as reliability demography insurance andmedicine They are defined by
119861 (120588) =
1
1205881205831015840
1
int
119902
0
119909119891 (119909) 119889119909
119871 (120588) =
1
1205831015840
1
int
119902
0
119909119891 (119909) 119889119909
(36)
respectively where 119902 = 119876BGHN(120588) = 119876119861(]) and ] =
2Φ[(120588120579)120572
] minus 1 (Section 2) for a given probability 120588 Fromint
119902
0
119909119891(119909)119889119909 = 1205831015840
1minus 119879(120588) we obtain 119861(120588) = 120588
minus1
[1 minus 119879(120588)1205831015840
1]
and 119871(120588) = 1 minus 119879(120588)1205831015840
1
33 Renyi Entropy The Renyi information of order 120585 for acontinuous random variable with density function 119891(119909) isdefined as
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (37)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 Applicationsof the Renyi entropy can be found in several areas suchas physics information theory and engineering to describe
Journal of Probability and Statistics 5
many nonlinear dynamical or chaotic systems [8] and instatistics as certain appropriately scaled test statistics (relativeRenyi information) for testing hypotheses in parametricmodels [9] Renyi [10] generalized the concept of informationtheory which allows for different averaging of probabilitiesvia 120585
For the BGHN distribution (4) the Renyi entropy isdefined by
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (38)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 From (4) we have
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
120585(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120585(119887minus1)
119889119909
(39)
For |119911| lt 1 and 119887 is a real noninteger the power series holds
(1 minus 119911)119887minus1
=
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119911
119895
(40)
where the binomial coefficient is defined for any real Using(40) in (39) twice 119868(120585) can be expressed as
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times
infin
sum
119895119896=0
(minus1)119895+119896
2119895
(
120585 (119886 minus 1) + 1
119895)
times (
120585 (119887 minus 1) + 119895 + 1
119896)
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
Φ[(
119909
120579
)
120572
]
119896
119889119909
(41)
Substituting Φ(119909) by the error function and setting 119906 =
(119909120579)120572 119868(120585) reduces to
119868 (120585) = 120572120585minus1
1205791minus120585
2120585(119887minus1)
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times int
infin
0
119906((120585(120572minus1)+1)120572)minus1
119890minus(1205852)119906
2
[erf ( 119906
radic2
)]
119897
119889119906
(42)
Following similar algebra that lead to (35) we obtain
119868 (120585) = 120572120585minus1
1205791minus120585
2[120585(119887minus1)+(120585(120572minus1)+1)2120572minus1]
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
(21198981+ 1) sdot sdot sdot (2119898
119897+ 1)119898
1 sdot sdot sdot 119898
119897
times 120585minus[1198981+sdotsdotsdot+119898
119897+1198972+(120585(120572minus1)+1)2120572]
times Γ(1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
)
(43)
Finally the Renyi entropy reduces to
120485119877(120585) = (1 minus 120585)
minus1
times
(120585 minus 1) log (120572) + (1 minus 120585) log (120579)
+ [120585 (119887 minus 1) +
120585 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120585 log(radic 2
120587
) + log[
[
infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)]
]
+ log[infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
times ((21198981+ 1) sdot sdot sdot (2119898
119897+ 1)
times 1198981 sdot sdot sdot 119898
119897)minus1
]
minus [1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
] log (120585)
+ log [Γ(1198981+sdot sdot sdot+119898
119897+
119897
2
+
120585 (120572minus1) + 1
2120572
)]
(44)
34 Reliability In the context of reliability the stress-strengthmodel describes the life of a component which has a randomstrength 119883
1that is subjected to a random stress 119883
2 The
component fails at the instant that the stress applied toit exceeds the strength and the component will functionsatisfactorily whenever 119883
1gt 119883
2 Hence 119877 = Pr(119883
2lt
1198831) is a measure of component reliability Here we derive
119877 when 1198831and 119883
2have independent BGHN(120572 120579 119886
1 1198871)
and BGHN(120572 120579 1198862 1198872) distributions with the same shape
parameters 120572 and 120579 The reliability 119877 becomes
119877 = int
infin
0
1198911(119909) 119865
2(119909) 119889119909 (45)
6 Journal of Probability and Statistics
where the cdf of 1198832and the density of 119883
1are obtained from
(6) and (10) as
1198652(119909) =
infin
sum
119895=0
119908119895(119886
2 1198872) 2Φ [(
119909
120579
)
120572
] minus 1
1198862+119895
1198911(119909) = radic
2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times
infin
sum
119903=0
119887119903(119886
1 1198871) 2Φ [(
119909
120579
)
120572
] minus 1
119903
(46)
respectively where
119908119895(119886
2 1198872) =
(minus1)119895
119861 (1198862 1198872)
(
1198872minus 1
119895)
119887119903(119886
1 1198871) =
infin
sum
119895=0
(minus1)119895
119861 (1198861 1198871)
(
1198871minus 1
119895) 119904
119903(119886
1+ 119895 minus 1)
(47)
refer to1198832and119883
1 respectively Hence
119877 = 120572radic2
120587
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
times int
infin
0
119909minus1
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
1198862+119895+119903
119889119909
(48)
Setting 119906 = 2Φ[(119909120579)120572
] minus 1 in the last integral the reli-ability of119883 reduces to
119877 =
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
1198862+ 119895 + 119903 + 1
(49)
4 Computational Issues
Here we show the practical use of (24) (26) (30) (35) and(49) If we truncate these infinite power series we have asimple way to compute the quantile function moments mgfmean deviations and reliability of the BGHN distributionThe question is how large the truncation limit should be
We now provide evidence that each infinite summationcan be truncated at twenty to yield sufficient accuracy Let119863
1
denote the absolute difference between the integrated version
1
119861 (119886 119887)
int
2Φ[(119909120579)120572]minus1
0
119905119886minus1
(1 minus 119905)119887minus1
119889119905 = 119880 (50)
where 119880 sim 119880(0 1) is a uniform variate on the unit intervaland the truncated version of (24) averaged over (all with step001) 119909 = 001 5 119886 = 001 10 119887 = 001 10
120572 = 001 10 and 120579 = 001 10 Let 1198632denote the
absolute difference between integrated version
1205831015840
119904=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
119909119904
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(51)
and the truncated version of (26) averaged over 119909 =
001 5 119886 = 001 10 119887 = 001 10120572 = 001 10
and 120579 = 001 10 Let 1198633denote the absolute difference
between the truncated version of (30) and the integratedversion
119872120572120579119886119887
(119905)
=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
(
120572
119909
)(
119909
120579
)
120572
exp [119905119909 minus 1
2
(
119909
120579
)
2120572
]
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(52)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198634denote
the absolute difference between the truncated version of (35)and the integrated version
119879 (119902) =
120572 2119887minus1
119861 (119886 119887)
radic2
120587
int
119902
0
(
119909
120579
)
120572
times exp [minus12
(
119909
120579
)
2120572
] 2Φ[(
119909
120579
)
120572
]minus1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(53)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198635denote
the absolute difference between the truncated version of (49)and the integrated version (45) averaged over 119909 = 001 5119886 = 001 10 119887 = 001 10 120572 = 001 10 and 120579 =
001 10We obtain the following estimates after extensive compu-
tations1198631= 231times10
minus201198632= 847times10
minus181198633= 122times10
minus211198634= 151 times 10
minus22 and1198635= 941 times 10
minus19 These estimates aresmall enough to suggest that the truncated versions of (24)(26) (30) (35) and (49) are reasonable practical use
It would be important to verify that each (untruncated)infinite series (such as (24) (26) (30) (35) and (49)) is con-vergent and provide valid values for its arguments Howeverthis will be a difficult mathematical problem that could beinvestigated in future work
Journal of Probability and Statistics 7
5 Properties of the BGHN Order Statistics
Order statistics have been used in a wide range of prob-lems including robust statistical estimation and detectionof outliers characterization of probability distributions andgoodness-of-fit tests entropy estimation analysis of censoredsamples reliability analysis quality control and strength ofmaterials
51 Mixture Form Suppose 1198831 119883
119899is a random sample
of size 119899 from a continuous distribution and let1198831119899
lt 1198832119899
lt
sdot sdot sdot lt 119883119899119899
denote the corresponding order statistics Therehas been a large amount of work relating tomoments of orderstatistics119883
119894119899 See Arnold et al [11] David and Nagaraja [12]
and Ahsanullah and Nevzorov [13] for excellent accounts Itis well known that the density function of119883
119894119899is given by
119891119894119899(119909) =
119891 (119909)
119861 (119894 119899 minus 119894 + 1)
119865(119909)119894minus1
[1 minus 119865 (119909)]119899minus119894
(54)
For the BGHN distribution Pescim et al [3] obtained
119891119894119899(119909) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119891119896119872119896(119909) (55)
where 119872119896denotes a sequence (119898
1 119898
119894+119896minus1) of 119894 + 119896 minus 1
nonnegative integers 119891119896119872119896
(119909) denotes the BGHN(120572 120579 (119894 +119896)119886 + sum
119894+119896minus1
119895=1119898119895 119887) density function defined under 119872
119896and
the constants 120578119896119872119896
are given by
120578119896119872119896
=
(minus1)119896+sum119894+119896minus1
119895=1119898119895(119899minus119894
119896) 119861 (119886 (119894 + 119896)+sum
119894+119896minus1
119895=1119898119895 119887) Γ(119887)
119894+119896minus1
119861(119886 119887)119894+119896
119861 (119894 119899minus 119894+1)prod119894+119896minus1
119895=1Γ (119887minus119898
119895)119898
119895 (119886+119898
119895)
(56)
The quantities 120578119896119872119896
are easily obtained given 119896 and asequence 119872
119896of indices 119898
1 119898
119894+119896minus1 The sums in (55)
extend over all (119894 + 119896)-tuples (1198961198981 119898
119894+119896minus1) and can be
implementable on a computer If 119887 gt 0 is an integer (55)holds but the indices 119898
1 119898
119894+119896minus1vary from zero to 119887 minus 1
Equation (55) reveals that the density function of the BGHNorder statistics can be expressed as an infinite mixture ofBGHN densities
52 Moments of Order Statistics Themoments of the BGHNorder statistics can be written directly in terms of themoments of BGHNdistributions from themixture form (55)We have
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119864 (119883119904
119894119896) (57)
where119883119894119896sim BGHN(120572 120579 (119894 + 119896)119886 + sum119894+119896minus1
119895=1119898119895 119887) and 119864(119883119904
119894119896)
can be determined from (26) Inserting (26) in (57) andchanging indices we can write
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
infin
sum
119902=0
119886⋆V
119904119902
(119902 + 119886⋆)
(58)
where
119886⋆
= (119894 + 119896) 119886 +
119894+119896minus1
sum
119895=1
119898119895 (59)
The moments 119864(119883119904
119894119899) can be determined based on the
explicit sum (58) using the software Mathematica They arealso computed from (55) by numerical integration using thestatistical software R [14] The values from both techniquesare usually close wheninfin is replaced by a moderate numbersuch as 20 in (58) For some values 119886 = 2 119887 = 3 and 119899 = 5Table 1 lists the numerical values for the moments 119864(119883119904
119894119899)
(119904 = 1 4 119894 = 1 5) and for the variance skewness andkurtosis
53 Generating Function The mgf of the BGHN orderstatistics can be written directly in terms of the BGHNmgf rsquosfrom the mixture form (55) and (30) We obtain
119872119894119899(119905) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119872120572120579119886⋆119887(119905) (60)
where119872120572120579119886⋆119887(119905) is the mgf of the BGHN(120572 120579 119886⋆ 119887) distri-
bution obtained from (30)
54 Mean Deviations The mean deviations of the orderstatistics in relation to the mean and the median are definedby
1205751(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 120583
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
1205752(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 119872
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
(61)
respectively where 120583119894= 119864(119883
119894119899) and 119872
119894= Median(119883
119894119899)
denotes the median Here 119872119894is obtained as the solution of
the nonlinear equation119899
sum
119903=119894
(
119899
119903) 119868
2Φ[(119872119894120579)120572]minus1
(119886 119887)
119903
times 1 minus 1198682Φ[(119872
119894120579)120572]minus1
(119886 119887)
119899minus119903
=
1
2
(62)
The measures 1205751(119883
119894119899) and 120575
2(119883
119894119899) follow from
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(63)
where 119869119894(119902) = int
infin
119902
119909119891119894119899(119909)119889119909 Using (55) we have
119869119894(119902) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119894119872119896
119879 (119902) (64)
where 119879(119902) is obtained from (35) using 119886⋆ given by (59) and
119887⋆
119903= 119887
119903(119886
⋆
119887) =
1
119861 (119886⋆ 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886
⋆
+ 119895 minus 1)
(65)
8 Journal of Probability and Statistics
Table 1 Moments of the BGHN order statistics for 120579 = 85 120572 = 3 119886 = 2 and 119887 = 3
Order statistics rarr 1198831 5
1198832 5
1198833 5
1198834 5
1198835 5
119904 = 1 267942 581946 473975 171571 023289119904 = 2 1810007 3931172 3201807 1159006 157328119904 = 3 1278683 2777184 2261923 8187821 1111451119904 = 4 938314 2037933 1659828 6008327 8155966Variance 1092078 544561 955281 864636 151904Skewness 057767 minus113596 minus054601 127135 536291Kurtosis 161751 406406 189737 291095 3107642
where
119904119903(119886
⋆
+ 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886⋆+ 119895
(
119886⋆
+ 119895 minus 1
119896)(
119896
119903) (66)
Bonferroni and Lorenz curves of the order statistics aregiven by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(67)
respectively where 119902 = 119865minus1
119894119899(120588) for a given probability 120588 From
int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus 119869
119894(120588) we obtain
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(68)
55 Renyi Entropy The Renyi entropy of the order statisticsis defined by
120485119877(120582) =
1
1 minus 120582
log [119867 (120582)] (69)
where 119867(120582) = int119891120582
119894119899(119909)119889119909 120582 gt 0 and 120582 = 1 From (54) it
follows that
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
int
infin
0
119891120582
(119909)
times [119865 (119909)]120582(119894minus1)
[1minus119865 (119909)]120582(119899minus119894)
119889119909
(70)
Using (40) in (70) we obtain
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
infin
sum
1198961=0
(minus1)1198961(
120582 (119894 minus 1)
1198961
)
times int
infin
0
119891120582
(119909) [119865 (119909)]120582(119894minus1)+119896
1119889119909
(71)
For 120582 gt 0 and 120582 = 1 we can expand [119865(119909)]120582(119894minus1)+1198961 as
119865(119909)120582(119894minus1)+119896
1= 1 minus [1 minus 119865 (119909)]
120582(119894minus1)+1198961
=
infin
sum
1199011=0
(minus1)1199011(
120582 (119894 minus 1) + 1198961+ 1
1199011
) [1 minus 119865 (119909)]1199011
(72)
and then
119865(119909)120582(119894minus1)+119896
1=
infin
sum
1199011=0
1199011
sum
1198971=0
(minus1)1199011+1198971
times (
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)119865(119909)1198971
(73)
Hence from (70) we can write
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
(minus1)1198961+1199011+1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
times int
infin
0
119891120582
(119909) 119865(119909)1198971119889119909
(74)
By replacing 119891(119909) and 119865(119909) by (10) and (16) given by Pescimet al [3] we obtain
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
[Γ (119887)]1198971(minus1)
1198961+1199011+1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 5
many nonlinear dynamical or chaotic systems [8] and instatistics as certain appropriately scaled test statistics (relativeRenyi information) for testing hypotheses in parametricmodels [9] Renyi [10] generalized the concept of informationtheory which allows for different averaging of probabilitiesvia 120585
For the BGHN distribution (4) the Renyi entropy isdefined by
120485119877(120585) =
1
1 minus 120585
log [119868 (120585)] (38)
where 119868(120585) = int119891120585
(119909)119889119909 120585 gt 0 and 120585 = 1 From (4) we have
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
120585(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120585(119887minus1)
119889119909
(39)
For |119911| lt 1 and 119887 is a real noninteger the power series holds
(1 minus 119911)119887minus1
=
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119911
119895
(40)
where the binomial coefficient is defined for any real Using(40) in (39) twice 119868(120585) can be expressed as
119868 (120585) =
2120585(119887minus1)
(120572radic2120587)
120585
[119861(119886 119887)]120585
times
infin
sum
119895119896=0
(minus1)119895+119896
2119895
(
120585 (119886 minus 1) + 1
119895)
times (
120585 (119887 minus 1) + 119895 + 1
119896)
times int
infin
0
119909minus120585
(
119909
120579
)
120572120585
119890minus(1205852)(119909120579)
2120572
Φ[(
119909
120579
)
120572
]
119896
119889119909
(41)
Substituting Φ(119909) by the error function and setting 119906 =
(119909120579)120572 119868(120585) reduces to
119868 (120585) = 120572120585minus1
1205791minus120585
2120585(119887minus1)
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times int
infin
0
119906((120585(120572minus1)+1)120572)minus1
119890minus(1205852)119906
2
[erf ( 119906
radic2
)]
119897
119889119906
(42)
Following similar algebra that lead to (35) we obtain
119868 (120585) = 120572120585minus1
1205791minus120585
2[120585(119887minus1)+(120585(120572minus1)+1)2120572minus1]
times (radic2
120587
)
120585infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
(21198981+ 1) sdot sdot sdot (2119898
119897+ 1)119898
1 sdot sdot sdot 119898
119897
times 120585minus[1198981+sdotsdotsdot+119898
119897+1198972+(120585(120572minus1)+1)2120572]
times Γ(1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
)
(43)
Finally the Renyi entropy reduces to
120485119877(120585) = (1 minus 120585)
minus1
times
(120585 minus 1) log (120572) + (1 minus 120585) log (120579)
+ [120585 (119887 minus 1) +
120585 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120585 log(radic 2
120587
) + log[
[
infin
sum
119895119896=0
119896
sum
119897=0
119908119895119896119897
(119886 119887)]
]
+ log[infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119897=0
(minus1)1198981+sdotsdotsdot+119898
119897
times ((21198981+ 1) sdot sdot sdot (2119898
119897+ 1)
times 1198981 sdot sdot sdot 119898
119897)minus1
]
minus [1198981+ sdot sdot sdot + 119898
119897+
119897
2
+
120585 (120572 minus 1) + 1
2120572
] log (120585)
+ log [Γ(1198981+sdot sdot sdot+119898
119897+
119897
2
+
120585 (120572minus1) + 1
2120572
)]
(44)
34 Reliability In the context of reliability the stress-strengthmodel describes the life of a component which has a randomstrength 119883
1that is subjected to a random stress 119883
2 The
component fails at the instant that the stress applied toit exceeds the strength and the component will functionsatisfactorily whenever 119883
1gt 119883
2 Hence 119877 = Pr(119883
2lt
1198831) is a measure of component reliability Here we derive
119877 when 1198831and 119883
2have independent BGHN(120572 120579 119886
1 1198871)
and BGHN(120572 120579 1198862 1198872) distributions with the same shape
parameters 120572 and 120579 The reliability 119877 becomes
119877 = int
infin
0
1198911(119909) 119865
2(119909) 119889119909 (45)
6 Journal of Probability and Statistics
where the cdf of 1198832and the density of 119883
1are obtained from
(6) and (10) as
1198652(119909) =
infin
sum
119895=0
119908119895(119886
2 1198872) 2Φ [(
119909
120579
)
120572
] minus 1
1198862+119895
1198911(119909) = radic
2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times
infin
sum
119903=0
119887119903(119886
1 1198871) 2Φ [(
119909
120579
)
120572
] minus 1
119903
(46)
respectively where
119908119895(119886
2 1198872) =
(minus1)119895
119861 (1198862 1198872)
(
1198872minus 1
119895)
119887119903(119886
1 1198871) =
infin
sum
119895=0
(minus1)119895
119861 (1198861 1198871)
(
1198871minus 1
119895) 119904
119903(119886
1+ 119895 minus 1)
(47)
refer to1198832and119883
1 respectively Hence
119877 = 120572radic2
120587
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
times int
infin
0
119909minus1
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
1198862+119895+119903
119889119909
(48)
Setting 119906 = 2Φ[(119909120579)120572
] minus 1 in the last integral the reli-ability of119883 reduces to
119877 =
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
1198862+ 119895 + 119903 + 1
(49)
4 Computational Issues
Here we show the practical use of (24) (26) (30) (35) and(49) If we truncate these infinite power series we have asimple way to compute the quantile function moments mgfmean deviations and reliability of the BGHN distributionThe question is how large the truncation limit should be
We now provide evidence that each infinite summationcan be truncated at twenty to yield sufficient accuracy Let119863
1
denote the absolute difference between the integrated version
1
119861 (119886 119887)
int
2Φ[(119909120579)120572]minus1
0
119905119886minus1
(1 minus 119905)119887minus1
119889119905 = 119880 (50)
where 119880 sim 119880(0 1) is a uniform variate on the unit intervaland the truncated version of (24) averaged over (all with step001) 119909 = 001 5 119886 = 001 10 119887 = 001 10
120572 = 001 10 and 120579 = 001 10 Let 1198632denote the
absolute difference between integrated version
1205831015840
119904=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
119909119904
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(51)
and the truncated version of (26) averaged over 119909 =
001 5 119886 = 001 10 119887 = 001 10120572 = 001 10
and 120579 = 001 10 Let 1198633denote the absolute difference
between the truncated version of (30) and the integratedversion
119872120572120579119886119887
(119905)
=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
(
120572
119909
)(
119909
120579
)
120572
exp [119905119909 minus 1
2
(
119909
120579
)
2120572
]
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(52)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198634denote
the absolute difference between the truncated version of (35)and the integrated version
119879 (119902) =
120572 2119887minus1
119861 (119886 119887)
radic2
120587
int
119902
0
(
119909
120579
)
120572
times exp [minus12
(
119909
120579
)
2120572
] 2Φ[(
119909
120579
)
120572
]minus1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(53)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198635denote
the absolute difference between the truncated version of (49)and the integrated version (45) averaged over 119909 = 001 5119886 = 001 10 119887 = 001 10 120572 = 001 10 and 120579 =
001 10We obtain the following estimates after extensive compu-
tations1198631= 231times10
minus201198632= 847times10
minus181198633= 122times10
minus211198634= 151 times 10
minus22 and1198635= 941 times 10
minus19 These estimates aresmall enough to suggest that the truncated versions of (24)(26) (30) (35) and (49) are reasonable practical use
It would be important to verify that each (untruncated)infinite series (such as (24) (26) (30) (35) and (49)) is con-vergent and provide valid values for its arguments Howeverthis will be a difficult mathematical problem that could beinvestigated in future work
Journal of Probability and Statistics 7
5 Properties of the BGHN Order Statistics
Order statistics have been used in a wide range of prob-lems including robust statistical estimation and detectionof outliers characterization of probability distributions andgoodness-of-fit tests entropy estimation analysis of censoredsamples reliability analysis quality control and strength ofmaterials
51 Mixture Form Suppose 1198831 119883
119899is a random sample
of size 119899 from a continuous distribution and let1198831119899
lt 1198832119899
lt
sdot sdot sdot lt 119883119899119899
denote the corresponding order statistics Therehas been a large amount of work relating tomoments of orderstatistics119883
119894119899 See Arnold et al [11] David and Nagaraja [12]
and Ahsanullah and Nevzorov [13] for excellent accounts Itis well known that the density function of119883
119894119899is given by
119891119894119899(119909) =
119891 (119909)
119861 (119894 119899 minus 119894 + 1)
119865(119909)119894minus1
[1 minus 119865 (119909)]119899minus119894
(54)
For the BGHN distribution Pescim et al [3] obtained
119891119894119899(119909) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119891119896119872119896(119909) (55)
where 119872119896denotes a sequence (119898
1 119898
119894+119896minus1) of 119894 + 119896 minus 1
nonnegative integers 119891119896119872119896
(119909) denotes the BGHN(120572 120579 (119894 +119896)119886 + sum
119894+119896minus1
119895=1119898119895 119887) density function defined under 119872
119896and
the constants 120578119896119872119896
are given by
120578119896119872119896
=
(minus1)119896+sum119894+119896minus1
119895=1119898119895(119899minus119894
119896) 119861 (119886 (119894 + 119896)+sum
119894+119896minus1
119895=1119898119895 119887) Γ(119887)
119894+119896minus1
119861(119886 119887)119894+119896
119861 (119894 119899minus 119894+1)prod119894+119896minus1
119895=1Γ (119887minus119898
119895)119898
119895 (119886+119898
119895)
(56)
The quantities 120578119896119872119896
are easily obtained given 119896 and asequence 119872
119896of indices 119898
1 119898
119894+119896minus1 The sums in (55)
extend over all (119894 + 119896)-tuples (1198961198981 119898
119894+119896minus1) and can be
implementable on a computer If 119887 gt 0 is an integer (55)holds but the indices 119898
1 119898
119894+119896minus1vary from zero to 119887 minus 1
Equation (55) reveals that the density function of the BGHNorder statistics can be expressed as an infinite mixture ofBGHN densities
52 Moments of Order Statistics Themoments of the BGHNorder statistics can be written directly in terms of themoments of BGHNdistributions from themixture form (55)We have
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119864 (119883119904
119894119896) (57)
where119883119894119896sim BGHN(120572 120579 (119894 + 119896)119886 + sum119894+119896minus1
119895=1119898119895 119887) and 119864(119883119904
119894119896)
can be determined from (26) Inserting (26) in (57) andchanging indices we can write
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
infin
sum
119902=0
119886⋆V
119904119902
(119902 + 119886⋆)
(58)
where
119886⋆
= (119894 + 119896) 119886 +
119894+119896minus1
sum
119895=1
119898119895 (59)
The moments 119864(119883119904
119894119899) can be determined based on the
explicit sum (58) using the software Mathematica They arealso computed from (55) by numerical integration using thestatistical software R [14] The values from both techniquesare usually close wheninfin is replaced by a moderate numbersuch as 20 in (58) For some values 119886 = 2 119887 = 3 and 119899 = 5Table 1 lists the numerical values for the moments 119864(119883119904
119894119899)
(119904 = 1 4 119894 = 1 5) and for the variance skewness andkurtosis
53 Generating Function The mgf of the BGHN orderstatistics can be written directly in terms of the BGHNmgf rsquosfrom the mixture form (55) and (30) We obtain
119872119894119899(119905) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119872120572120579119886⋆119887(119905) (60)
where119872120572120579119886⋆119887(119905) is the mgf of the BGHN(120572 120579 119886⋆ 119887) distri-
bution obtained from (30)
54 Mean Deviations The mean deviations of the orderstatistics in relation to the mean and the median are definedby
1205751(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 120583
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
1205752(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 119872
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
(61)
respectively where 120583119894= 119864(119883
119894119899) and 119872
119894= Median(119883
119894119899)
denotes the median Here 119872119894is obtained as the solution of
the nonlinear equation119899
sum
119903=119894
(
119899
119903) 119868
2Φ[(119872119894120579)120572]minus1
(119886 119887)
119903
times 1 minus 1198682Φ[(119872
119894120579)120572]minus1
(119886 119887)
119899minus119903
=
1
2
(62)
The measures 1205751(119883
119894119899) and 120575
2(119883
119894119899) follow from
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(63)
where 119869119894(119902) = int
infin
119902
119909119891119894119899(119909)119889119909 Using (55) we have
119869119894(119902) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119894119872119896
119879 (119902) (64)
where 119879(119902) is obtained from (35) using 119886⋆ given by (59) and
119887⋆
119903= 119887
119903(119886
⋆
119887) =
1
119861 (119886⋆ 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886
⋆
+ 119895 minus 1)
(65)
8 Journal of Probability and Statistics
Table 1 Moments of the BGHN order statistics for 120579 = 85 120572 = 3 119886 = 2 and 119887 = 3
Order statistics rarr 1198831 5
1198832 5
1198833 5
1198834 5
1198835 5
119904 = 1 267942 581946 473975 171571 023289119904 = 2 1810007 3931172 3201807 1159006 157328119904 = 3 1278683 2777184 2261923 8187821 1111451119904 = 4 938314 2037933 1659828 6008327 8155966Variance 1092078 544561 955281 864636 151904Skewness 057767 minus113596 minus054601 127135 536291Kurtosis 161751 406406 189737 291095 3107642
where
119904119903(119886
⋆
+ 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886⋆+ 119895
(
119886⋆
+ 119895 minus 1
119896)(
119896
119903) (66)
Bonferroni and Lorenz curves of the order statistics aregiven by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(67)
respectively where 119902 = 119865minus1
119894119899(120588) for a given probability 120588 From
int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus 119869
119894(120588) we obtain
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(68)
55 Renyi Entropy The Renyi entropy of the order statisticsis defined by
120485119877(120582) =
1
1 minus 120582
log [119867 (120582)] (69)
where 119867(120582) = int119891120582
119894119899(119909)119889119909 120582 gt 0 and 120582 = 1 From (54) it
follows that
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
int
infin
0
119891120582
(119909)
times [119865 (119909)]120582(119894minus1)
[1minus119865 (119909)]120582(119899minus119894)
119889119909
(70)
Using (40) in (70) we obtain
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
infin
sum
1198961=0
(minus1)1198961(
120582 (119894 minus 1)
1198961
)
times int
infin
0
119891120582
(119909) [119865 (119909)]120582(119894minus1)+119896
1119889119909
(71)
For 120582 gt 0 and 120582 = 1 we can expand [119865(119909)]120582(119894minus1)+1198961 as
119865(119909)120582(119894minus1)+119896
1= 1 minus [1 minus 119865 (119909)]
120582(119894minus1)+1198961
=
infin
sum
1199011=0
(minus1)1199011(
120582 (119894 minus 1) + 1198961+ 1
1199011
) [1 minus 119865 (119909)]1199011
(72)
and then
119865(119909)120582(119894minus1)+119896
1=
infin
sum
1199011=0
1199011
sum
1198971=0
(minus1)1199011+1198971
times (
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)119865(119909)1198971
(73)
Hence from (70) we can write
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
(minus1)1198961+1199011+1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
times int
infin
0
119891120582
(119909) 119865(119909)1198971119889119909
(74)
By replacing 119891(119909) and 119865(119909) by (10) and (16) given by Pescimet al [3] we obtain
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
[Γ (119887)]1198971(minus1)
1198961+1199011+1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Probability and Statistics
where the cdf of 1198832and the density of 119883
1are obtained from
(6) and (10) as
1198652(119909) =
infin
sum
119895=0
119908119895(119886
2 1198872) 2Φ [(
119909
120579
)
120572
] minus 1
1198862+119895
1198911(119909) = radic
2
120587
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times
infin
sum
119903=0
119887119903(119886
1 1198871) 2Φ [(
119909
120579
)
120572
] minus 1
119903
(46)
respectively where
119908119895(119886
2 1198872) =
(minus1)119895
119861 (1198862 1198872)
(
1198872minus 1
119895)
119887119903(119886
1 1198871) =
infin
sum
119895=0
(minus1)119895
119861 (1198861 1198871)
(
1198871minus 1
119895) 119904
119903(119886
1+ 119895 minus 1)
(47)
refer to1198832and119883
1 respectively Hence
119877 = 120572radic2
120587
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
times int
infin
0
119909minus1
(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
2Φ[(
119909
120579
)
120572
] minus 1
1198862+119895+119903
119889119909
(48)
Setting 119906 = 2Φ[(119909120579)120572
] minus 1 in the last integral the reli-ability of119883 reduces to
119877 =
infin
sum
119895119903=0
119908119895(119886
2 1198872) 119887
119903(119886
1 1198871)
1198862+ 119895 + 119903 + 1
(49)
4 Computational Issues
Here we show the practical use of (24) (26) (30) (35) and(49) If we truncate these infinite power series we have asimple way to compute the quantile function moments mgfmean deviations and reliability of the BGHN distributionThe question is how large the truncation limit should be
We now provide evidence that each infinite summationcan be truncated at twenty to yield sufficient accuracy Let119863
1
denote the absolute difference between the integrated version
1
119861 (119886 119887)
int
2Φ[(119909120579)120572]minus1
0
119905119886minus1
(1 minus 119905)119887minus1
119889119905 = 119880 (50)
where 119880 sim 119880(0 1) is a uniform variate on the unit intervaland the truncated version of (24) averaged over (all with step001) 119909 = 001 5 119886 = 001 10 119887 = 001 10
120572 = 001 10 and 120579 = 001 10 Let 1198632denote the
absolute difference between integrated version
1205831015840
119904=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
119909119904
(
120572
119909
)(
119909
120579
)
120572
119890minus(12)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(51)
and the truncated version of (26) averaged over 119909 =
001 5 119886 = 001 10 119887 = 001 10120572 = 001 10
and 120579 = 001 10 Let 1198633denote the absolute difference
between the truncated version of (30) and the integratedversion
119872120572120579119886119887
(119905)
=
2119887minus1
119861 (119886 119887)
radic2
120587
int
infin
0
(
120572
119909
)(
119909
120579
)
120572
exp [119905119909 minus 1
2
(
119909
120579
)
2120572
]
times 2Φ[(
119909
120579
)
120572
] minus 1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(52)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198634denote
the absolute difference between the truncated version of (35)and the integrated version
119879 (119902) =
120572 2119887minus1
119861 (119886 119887)
radic2
120587
int
119902
0
(
119909
120579
)
120572
times exp [minus12
(
119909
120579
)
2120572
] 2Φ[(
119909
120579
)
120572
]minus1
119886minus1
times 1 minus Φ[(
119909
120579
)
120572
]
119887minus1
119889119909
(53)
averaged over 119909 = 001 5 119886 = 001 10 119887 = 001
10 120572 = 001 10 and 120579 = 001 10 Let 1198635denote
the absolute difference between the truncated version of (49)and the integrated version (45) averaged over 119909 = 001 5119886 = 001 10 119887 = 001 10 120572 = 001 10 and 120579 =
001 10We obtain the following estimates after extensive compu-
tations1198631= 231times10
minus201198632= 847times10
minus181198633= 122times10
minus211198634= 151 times 10
minus22 and1198635= 941 times 10
minus19 These estimates aresmall enough to suggest that the truncated versions of (24)(26) (30) (35) and (49) are reasonable practical use
It would be important to verify that each (untruncated)infinite series (such as (24) (26) (30) (35) and (49)) is con-vergent and provide valid values for its arguments Howeverthis will be a difficult mathematical problem that could beinvestigated in future work
Journal of Probability and Statistics 7
5 Properties of the BGHN Order Statistics
Order statistics have been used in a wide range of prob-lems including robust statistical estimation and detectionof outliers characterization of probability distributions andgoodness-of-fit tests entropy estimation analysis of censoredsamples reliability analysis quality control and strength ofmaterials
51 Mixture Form Suppose 1198831 119883
119899is a random sample
of size 119899 from a continuous distribution and let1198831119899
lt 1198832119899
lt
sdot sdot sdot lt 119883119899119899
denote the corresponding order statistics Therehas been a large amount of work relating tomoments of orderstatistics119883
119894119899 See Arnold et al [11] David and Nagaraja [12]
and Ahsanullah and Nevzorov [13] for excellent accounts Itis well known that the density function of119883
119894119899is given by
119891119894119899(119909) =
119891 (119909)
119861 (119894 119899 minus 119894 + 1)
119865(119909)119894minus1
[1 minus 119865 (119909)]119899minus119894
(54)
For the BGHN distribution Pescim et al [3] obtained
119891119894119899(119909) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119891119896119872119896(119909) (55)
where 119872119896denotes a sequence (119898
1 119898
119894+119896minus1) of 119894 + 119896 minus 1
nonnegative integers 119891119896119872119896
(119909) denotes the BGHN(120572 120579 (119894 +119896)119886 + sum
119894+119896minus1
119895=1119898119895 119887) density function defined under 119872
119896and
the constants 120578119896119872119896
are given by
120578119896119872119896
=
(minus1)119896+sum119894+119896minus1
119895=1119898119895(119899minus119894
119896) 119861 (119886 (119894 + 119896)+sum
119894+119896minus1
119895=1119898119895 119887) Γ(119887)
119894+119896minus1
119861(119886 119887)119894+119896
119861 (119894 119899minus 119894+1)prod119894+119896minus1
119895=1Γ (119887minus119898
119895)119898
119895 (119886+119898
119895)
(56)
The quantities 120578119896119872119896
are easily obtained given 119896 and asequence 119872
119896of indices 119898
1 119898
119894+119896minus1 The sums in (55)
extend over all (119894 + 119896)-tuples (1198961198981 119898
119894+119896minus1) and can be
implementable on a computer If 119887 gt 0 is an integer (55)holds but the indices 119898
1 119898
119894+119896minus1vary from zero to 119887 minus 1
Equation (55) reveals that the density function of the BGHNorder statistics can be expressed as an infinite mixture ofBGHN densities
52 Moments of Order Statistics Themoments of the BGHNorder statistics can be written directly in terms of themoments of BGHNdistributions from themixture form (55)We have
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119864 (119883119904
119894119896) (57)
where119883119894119896sim BGHN(120572 120579 (119894 + 119896)119886 + sum119894+119896minus1
119895=1119898119895 119887) and 119864(119883119904
119894119896)
can be determined from (26) Inserting (26) in (57) andchanging indices we can write
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
infin
sum
119902=0
119886⋆V
119904119902
(119902 + 119886⋆)
(58)
where
119886⋆
= (119894 + 119896) 119886 +
119894+119896minus1
sum
119895=1
119898119895 (59)
The moments 119864(119883119904
119894119899) can be determined based on the
explicit sum (58) using the software Mathematica They arealso computed from (55) by numerical integration using thestatistical software R [14] The values from both techniquesare usually close wheninfin is replaced by a moderate numbersuch as 20 in (58) For some values 119886 = 2 119887 = 3 and 119899 = 5Table 1 lists the numerical values for the moments 119864(119883119904
119894119899)
(119904 = 1 4 119894 = 1 5) and for the variance skewness andkurtosis
53 Generating Function The mgf of the BGHN orderstatistics can be written directly in terms of the BGHNmgf rsquosfrom the mixture form (55) and (30) We obtain
119872119894119899(119905) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119872120572120579119886⋆119887(119905) (60)
where119872120572120579119886⋆119887(119905) is the mgf of the BGHN(120572 120579 119886⋆ 119887) distri-
bution obtained from (30)
54 Mean Deviations The mean deviations of the orderstatistics in relation to the mean and the median are definedby
1205751(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 120583
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
1205752(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 119872
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
(61)
respectively where 120583119894= 119864(119883
119894119899) and 119872
119894= Median(119883
119894119899)
denotes the median Here 119872119894is obtained as the solution of
the nonlinear equation119899
sum
119903=119894
(
119899
119903) 119868
2Φ[(119872119894120579)120572]minus1
(119886 119887)
119903
times 1 minus 1198682Φ[(119872
119894120579)120572]minus1
(119886 119887)
119899minus119903
=
1
2
(62)
The measures 1205751(119883
119894119899) and 120575
2(119883
119894119899) follow from
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(63)
where 119869119894(119902) = int
infin
119902
119909119891119894119899(119909)119889119909 Using (55) we have
119869119894(119902) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119894119872119896
119879 (119902) (64)
where 119879(119902) is obtained from (35) using 119886⋆ given by (59) and
119887⋆
119903= 119887
119903(119886
⋆
119887) =
1
119861 (119886⋆ 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886
⋆
+ 119895 minus 1)
(65)
8 Journal of Probability and Statistics
Table 1 Moments of the BGHN order statistics for 120579 = 85 120572 = 3 119886 = 2 and 119887 = 3
Order statistics rarr 1198831 5
1198832 5
1198833 5
1198834 5
1198835 5
119904 = 1 267942 581946 473975 171571 023289119904 = 2 1810007 3931172 3201807 1159006 157328119904 = 3 1278683 2777184 2261923 8187821 1111451119904 = 4 938314 2037933 1659828 6008327 8155966Variance 1092078 544561 955281 864636 151904Skewness 057767 minus113596 minus054601 127135 536291Kurtosis 161751 406406 189737 291095 3107642
where
119904119903(119886
⋆
+ 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886⋆+ 119895
(
119886⋆
+ 119895 minus 1
119896)(
119896
119903) (66)
Bonferroni and Lorenz curves of the order statistics aregiven by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(67)
respectively where 119902 = 119865minus1
119894119899(120588) for a given probability 120588 From
int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus 119869
119894(120588) we obtain
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(68)
55 Renyi Entropy The Renyi entropy of the order statisticsis defined by
120485119877(120582) =
1
1 minus 120582
log [119867 (120582)] (69)
where 119867(120582) = int119891120582
119894119899(119909)119889119909 120582 gt 0 and 120582 = 1 From (54) it
follows that
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
int
infin
0
119891120582
(119909)
times [119865 (119909)]120582(119894minus1)
[1minus119865 (119909)]120582(119899minus119894)
119889119909
(70)
Using (40) in (70) we obtain
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
infin
sum
1198961=0
(minus1)1198961(
120582 (119894 minus 1)
1198961
)
times int
infin
0
119891120582
(119909) [119865 (119909)]120582(119894minus1)+119896
1119889119909
(71)
For 120582 gt 0 and 120582 = 1 we can expand [119865(119909)]120582(119894minus1)+1198961 as
119865(119909)120582(119894minus1)+119896
1= 1 minus [1 minus 119865 (119909)]
120582(119894minus1)+1198961
=
infin
sum
1199011=0
(minus1)1199011(
120582 (119894 minus 1) + 1198961+ 1
1199011
) [1 minus 119865 (119909)]1199011
(72)
and then
119865(119909)120582(119894minus1)+119896
1=
infin
sum
1199011=0
1199011
sum
1198971=0
(minus1)1199011+1198971
times (
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)119865(119909)1198971
(73)
Hence from (70) we can write
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
(minus1)1198961+1199011+1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
times int
infin
0
119891120582
(119909) 119865(119909)1198971119889119909
(74)
By replacing 119891(119909) and 119865(119909) by (10) and (16) given by Pescimet al [3] we obtain
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
[Γ (119887)]1198971(minus1)
1198961+1199011+1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 7
5 Properties of the BGHN Order Statistics
Order statistics have been used in a wide range of prob-lems including robust statistical estimation and detectionof outliers characterization of probability distributions andgoodness-of-fit tests entropy estimation analysis of censoredsamples reliability analysis quality control and strength ofmaterials
51 Mixture Form Suppose 1198831 119883
119899is a random sample
of size 119899 from a continuous distribution and let1198831119899
lt 1198832119899
lt
sdot sdot sdot lt 119883119899119899
denote the corresponding order statistics Therehas been a large amount of work relating tomoments of orderstatistics119883
119894119899 See Arnold et al [11] David and Nagaraja [12]
and Ahsanullah and Nevzorov [13] for excellent accounts Itis well known that the density function of119883
119894119899is given by
119891119894119899(119909) =
119891 (119909)
119861 (119894 119899 minus 119894 + 1)
119865(119909)119894minus1
[1 minus 119865 (119909)]119899minus119894
(54)
For the BGHN distribution Pescim et al [3] obtained
119891119894119899(119909) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119891119896119872119896(119909) (55)
where 119872119896denotes a sequence (119898
1 119898
119894+119896minus1) of 119894 + 119896 minus 1
nonnegative integers 119891119896119872119896
(119909) denotes the BGHN(120572 120579 (119894 +119896)119886 + sum
119894+119896minus1
119895=1119898119895 119887) density function defined under 119872
119896and
the constants 120578119896119872119896
are given by
120578119896119872119896
=
(minus1)119896+sum119894+119896minus1
119895=1119898119895(119899minus119894
119896) 119861 (119886 (119894 + 119896)+sum
119894+119896minus1
119895=1119898119895 119887) Γ(119887)
119894+119896minus1
119861(119886 119887)119894+119896
119861 (119894 119899minus 119894+1)prod119894+119896minus1
119895=1Γ (119887minus119898
119895)119898
119895 (119886+119898
119895)
(56)
The quantities 120578119896119872119896
are easily obtained given 119896 and asequence 119872
119896of indices 119898
1 119898
119894+119896minus1 The sums in (55)
extend over all (119894 + 119896)-tuples (1198961198981 119898
119894+119896minus1) and can be
implementable on a computer If 119887 gt 0 is an integer (55)holds but the indices 119898
1 119898
119894+119896minus1vary from zero to 119887 minus 1
Equation (55) reveals that the density function of the BGHNorder statistics can be expressed as an infinite mixture ofBGHN densities
52 Moments of Order Statistics Themoments of the BGHNorder statistics can be written directly in terms of themoments of BGHNdistributions from themixture form (55)We have
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119864 (119883119904
119894119896) (57)
where119883119894119896sim BGHN(120572 120579 (119894 + 119896)119886 + sum119894+119896minus1
119895=1119898119895 119887) and 119864(119883119904
119894119896)
can be determined from (26) Inserting (26) in (57) andchanging indices we can write
119864 (119883119904
119894119899) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
infin
sum
119902=0
119886⋆V
119904119902
(119902 + 119886⋆)
(58)
where
119886⋆
= (119894 + 119896) 119886 +
119894+119896minus1
sum
119895=1
119898119895 (59)
The moments 119864(119883119904
119894119899) can be determined based on the
explicit sum (58) using the software Mathematica They arealso computed from (55) by numerical integration using thestatistical software R [14] The values from both techniquesare usually close wheninfin is replaced by a moderate numbersuch as 20 in (58) For some values 119886 = 2 119887 = 3 and 119899 = 5Table 1 lists the numerical values for the moments 119864(119883119904
119894119899)
(119904 = 1 4 119894 = 1 5) and for the variance skewness andkurtosis
53 Generating Function The mgf of the BGHN orderstatistics can be written directly in terms of the BGHNmgf rsquosfrom the mixture form (55) and (30) We obtain
119872119894119899(119905) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119896119872119896
119872120572120579119886⋆119887(119905) (60)
where119872120572120579119886⋆119887(119905) is the mgf of the BGHN(120572 120579 119886⋆ 119887) distri-
bution obtained from (30)
54 Mean Deviations The mean deviations of the orderstatistics in relation to the mean and the median are definedby
1205751(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 120583
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
1205752(119883
119894119899) = int
infin
0
1003816100381610038161003816119909 minus 119872
119894
1003816100381610038161003816119891119894119899(119909) 119889119909
(61)
respectively where 120583119894= 119864(119883
119894119899) and 119872
119894= Median(119883
119894119899)
denotes the median Here 119872119894is obtained as the solution of
the nonlinear equation119899
sum
119903=119894
(
119899
119903) 119868
2Φ[(119872119894120579)120572]minus1
(119886 119887)
119903
times 1 minus 1198682Φ[(119872
119894120579)120572]minus1
(119886 119887)
119899minus119903
=
1
2
(62)
The measures 1205751(119883
119894119899) and 120575
2(119883
119894119899) follow from
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(63)
where 119869119894(119902) = int
infin
119902
119909119891119894119899(119909)119889119909 Using (55) we have
119869119894(119902) =
119899minus119894
sum
119896=0
infin
sum
1198981=0
sdot sdot sdot
infin
sum
119898119894+119896minus1
=0
120578119894119872119896
119879 (119902) (64)
where 119879(119902) is obtained from (35) using 119886⋆ given by (59) and
119887⋆
119903= 119887
119903(119886
⋆
119887) =
1
119861 (119886⋆ 119887)
infin
sum
119895=0
(minus1)119895
(
119887 minus 1
119895) 119904
119903(119886
⋆
+ 119895 minus 1)
(65)
8 Journal of Probability and Statistics
Table 1 Moments of the BGHN order statistics for 120579 = 85 120572 = 3 119886 = 2 and 119887 = 3
Order statistics rarr 1198831 5
1198832 5
1198833 5
1198834 5
1198835 5
119904 = 1 267942 581946 473975 171571 023289119904 = 2 1810007 3931172 3201807 1159006 157328119904 = 3 1278683 2777184 2261923 8187821 1111451119904 = 4 938314 2037933 1659828 6008327 8155966Variance 1092078 544561 955281 864636 151904Skewness 057767 minus113596 minus054601 127135 536291Kurtosis 161751 406406 189737 291095 3107642
where
119904119903(119886
⋆
+ 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886⋆+ 119895
(
119886⋆
+ 119895 minus 1
119896)(
119896
119903) (66)
Bonferroni and Lorenz curves of the order statistics aregiven by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(67)
respectively where 119902 = 119865minus1
119894119899(120588) for a given probability 120588 From
int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus 119869
119894(120588) we obtain
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(68)
55 Renyi Entropy The Renyi entropy of the order statisticsis defined by
120485119877(120582) =
1
1 minus 120582
log [119867 (120582)] (69)
where 119867(120582) = int119891120582
119894119899(119909)119889119909 120582 gt 0 and 120582 = 1 From (54) it
follows that
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
int
infin
0
119891120582
(119909)
times [119865 (119909)]120582(119894minus1)
[1minus119865 (119909)]120582(119899minus119894)
119889119909
(70)
Using (40) in (70) we obtain
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
infin
sum
1198961=0
(minus1)1198961(
120582 (119894 minus 1)
1198961
)
times int
infin
0
119891120582
(119909) [119865 (119909)]120582(119894minus1)+119896
1119889119909
(71)
For 120582 gt 0 and 120582 = 1 we can expand [119865(119909)]120582(119894minus1)+1198961 as
119865(119909)120582(119894minus1)+119896
1= 1 minus [1 minus 119865 (119909)]
120582(119894minus1)+1198961
=
infin
sum
1199011=0
(minus1)1199011(
120582 (119894 minus 1) + 1198961+ 1
1199011
) [1 minus 119865 (119909)]1199011
(72)
and then
119865(119909)120582(119894minus1)+119896
1=
infin
sum
1199011=0
1199011
sum
1198971=0
(minus1)1199011+1198971
times (
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)119865(119909)1198971
(73)
Hence from (70) we can write
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
(minus1)1198961+1199011+1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
times int
infin
0
119891120582
(119909) 119865(119909)1198971119889119909
(74)
By replacing 119891(119909) and 119865(119909) by (10) and (16) given by Pescimet al [3] we obtain
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
[Γ (119887)]1198971(minus1)
1198961+1199011+1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Probability and Statistics
Table 1 Moments of the BGHN order statistics for 120579 = 85 120572 = 3 119886 = 2 and 119887 = 3
Order statistics rarr 1198831 5
1198832 5
1198833 5
1198834 5
1198835 5
119904 = 1 267942 581946 473975 171571 023289119904 = 2 1810007 3931172 3201807 1159006 157328119904 = 3 1278683 2777184 2261923 8187821 1111451119904 = 4 938314 2037933 1659828 6008327 8155966Variance 1092078 544561 955281 864636 151904Skewness 057767 minus113596 minus054601 127135 536291Kurtosis 161751 406406 189737 291095 3107642
where
119904119903(119886
⋆
+ 119895 minus 1) =
infin
sum
119896=119903
(minus1)119903+119896
119886⋆+ 119895
(
119886⋆
+ 119895 minus 1
119896)(
119896
119903) (66)
Bonferroni and Lorenz curves of the order statistics aregiven by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(67)
respectively where 119902 = 119865minus1
119894119899(120588) for a given probability 120588 From
int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus 119869
119894(120588) we obtain
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(68)
55 Renyi Entropy The Renyi entropy of the order statisticsis defined by
120485119877(120582) =
1
1 minus 120582
log [119867 (120582)] (69)
where 119867(120582) = int119891120582
119894119899(119909)119889119909 120582 gt 0 and 120582 = 1 From (54) it
follows that
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
int
infin
0
119891120582
(119909)
times [119865 (119909)]120582(119894minus1)
[1minus119865 (119909)]120582(119899minus119894)
119889119909
(70)
Using (40) in (70) we obtain
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
infin
sum
1198961=0
(minus1)1198961(
120582 (119894 minus 1)
1198961
)
times int
infin
0
119891120582
(119909) [119865 (119909)]120582(119894minus1)+119896
1119889119909
(71)
For 120582 gt 0 and 120582 = 1 we can expand [119865(119909)]120582(119894minus1)+1198961 as
119865(119909)120582(119894minus1)+119896
1= 1 minus [1 minus 119865 (119909)]
120582(119894minus1)+1198961
=
infin
sum
1199011=0
(minus1)1199011(
120582 (119894 minus 1) + 1198961+ 1
1199011
) [1 minus 119865 (119909)]1199011
(72)
and then
119865(119909)120582(119894minus1)+119896
1=
infin
sum
1199011=0
1199011
sum
1198971=0
(minus1)1199011+1198971
times (
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)119865(119909)1198971
(73)
Hence from (70) we can write
119867(120582) =
1
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
(minus1)1198961+1199011+1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
times int
infin
0
119891120582
(119909) 119865(119909)1198971119889119909
(74)
By replacing 119891(119909) and 119865(119909) by (10) and (16) given by Pescimet al [3] we obtain
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
[Γ (119887)]1198971(minus1)
1198961+1199011+1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 9
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
120582(119886minus1)
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
times[
[
infin
sum
119895=0
(minus1)119895
2Φ [(119909120579)120572
] minus 1119895
Γ (119887 minus 119895) 119895 (119886 + 119895)
]
]
1198971
119889119909
(75)
Using the identity (suminfin
119894=0119886119894)119896
= suminfin
1198981119898119896=0
1198861198981
sdot sdot sdot 119886119898119896
(for119896 positive integer) in (4) we have
119867(120582) =
2120582(119887minus1)
[120572 (radic2120587)]
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times int
infin
0
119909minus120582
(
119909
120579
)
120572120582
119890minus(1205822)(119909120579)
2120572
times 2Φ[(
119909
120579
)
120572
] minus 1
119886(120582+1198971)minus120582+sum
1198971
119895=1119898119895
times 1 minus Φ[(
119909
120579
)
120572
]
120582(119887minus1)
119889119909
(76)
where
119888119896111990111198971(119886 119887) =
(minus1)1198961+1199011+1198971[Γ (119887)]
1198971
[119861 (119886 119887)]1198971
times (
120582 (119894 minus 1)
1198961
)(
120582 (119894 minus 1) + 1198961+ 1
1199011
)(
1199011
1198971
)
(77)Using the power series (40) in (76) twice and replacing Φ(119909)by the error function erf(119909) (defined in Section 32) we obtainafter some algebra
119867(120582) =
120572120582minus1
1205791minus120582
2[120582(119887minus1)+(120582(120572minus1)+1)2120572minus1]
(radic2120587)
120582
[119861 (119894 119899 minus 119894 + 1)]120582
times
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ (119887 minus 119898
119895)119898
119895 (119886 + 119898
119895)
times
infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
times
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
(minus1)1198981+sdotsdotsdot+119898
1199041
(21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)1198981 sdot sdot sdot 119898
1199041
times 120582minus[1198981+sdotsdotsdot+119898
1199041+11990412+(120582(120572minus1)+1)2120572]
timesΓ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572minus1) +1
2120572
)
(78)where the quantity 119889
119903119904119905is well defined by Pescim et al [3]
(More details see equation (30) in [3])Finally the entropy of the order statistics can be expressed
as120485119877(120582) = (1 minus 120582)
minus1
times
(120582 minus 1) log (120572) + (1 minus 120582) log (120579)
+ [120582 (119887 minus 1) +
120582 (120572 minus 1) + 1
2120572
minus 1] log (2)
+ 120582 log(radic 2
120587
) minus 120582 log [119861 (119894 119899 minus 119894 + 1)]
+ log[
[
infin
sum
11989611199011=0
1199011
sum
1198971=0
119888119896111990111198971(119886 119887)
]
]
+log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981198971=0
(minus1)1198981+sdotsdotsdot+119898
1198971
prod1198971
119895=1Γ(119887minus119898
119895)119898
119895 (119886+119898
119895)
]
]
+ log(infin
sum
11990311199041=0
1199041
sum
V1=0
11988911990311199041V1
)
+ log[
[
infin
sum
1198981=0
sdot sdot sdot
infin
sum
1198981199041=0
((minus1)1198981+sdotsdotsdot+119898
1199041 )
times ( (21198981+ 1) sdot sdot sdot (2119898
1199041
+ 1)
times 1198981 sdot sdot sdot 119898
1199041
)
minus1
]
]
minus [1198981+ sdot sdot sdot + 119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
] log (120582)
+ log [Γ(1198981+sdot sdot sdot+119898
1199041
+
1199041
2
+
120582 (120572 minus 1) + 1
2120572
)]
(79)Equation (79) is the main result of this section
An alternative expansion for the density function of theorder statistics derived from the identity (20) was proposedby Pescim et al [3] A second density function representationand alternative expressions for some measures of the orderstatistics are given in Appendix A
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Probability and Statistics
6 Lifetime Analysis
Let 119883119894be a random variable having density function (4)
where 120596 = (120572 120579 119886 119887)119879 is the vector of parameters The data
encountered in survival analysis and reliability studies areoften censored A very simple random censoring mechanismthat is often realistic is one in which each individual 119894 isassumed to have a lifetime 119883
119894and a censoring time 119862
119894
where119883119894and 119862
119894are independent random variables Suppose
that the data consist of 119899 independent observations 119909119894=
min(119883119894 119862
119894) for 119894 = 1 119899 The distribution of 119862
119894does not
depend on any of the unknown parameters of the distributionof 119883
119894 Parametric inference for such data are usually based
on likelihood methods and their asymptotic theory Thecensored log-likelihood 119897(120596) for the model parameters is
119897 (120596) = 119903 log(radic 2
120587
) +sum
119894isin119865
log( 120572
119909119894
) + 120572sum
119894isin119865
log(119909119894
120579
)
minus
1
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+ 119903 (119887 minus 1) log (2) minus 119903 log [119861 (119886 119887)]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)
(80)
where 119903 is the number of failures and 119865 and 119862 denote theuncensored and censored sets of observations respectively
The score functions for the parameters 120572 120579 119886 and 119887 aregiven by
119880120572(120596) =
119903
2
+ sum
119894isin119865
log(119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894log (119909
119894120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880120579(120596) = minus119903 (
120572
120579
) + (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
timessum
119894isin119865
V119894(120572120579)
119875 (119909119894)
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus 2119887minus1
sum
119894isin119862
V119894(120572120579) [119875 (119909
119894)]119886minus1
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
119880119886(120596) = 119903 [120595 (119886 + 119887) minus 120595 (119886)] + sum
119894isin119865
log [119875 (119909119894)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119886
[1 minus 119876 (119909119894)]
119880119887(120596) = 119903 log (2) + 119903 [120595 (119886 + 119887) minus 120595 (119887)]
+ sum
119894isin119865
log 119863 (119909119894) minus sum
119894isin119862
119868119875(119909119894)(119886 119887)|
119887
[1 minus 119876 (119909119894)]
(81)
where
V119894=exp [minus1
2
(
119909119894
120579
)
2120572
] (
119909119894
120579
)
120572
119875 (119909119894)=2Φ [(
119909119894
120579
)
120572
]minus1
119863 (119909119894) = 1 minus Φ[(
119909119894
120579
)
120572
] 119876 (119909119894) = 119868
2Φ[(119909119894120579)120572]minus1
(119886 119887)
119868119875(119909119894)(119886 119887)|
119886=
120597
120597119886
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
119868119875(119909119894)(119886 119887)|
119887=
120597
120597119887
[
1
119861 (119886 119887)
int
119875(119909119894)
0
119908119886minus1
(1 minus 119908)119887minus1
119889119908]
(82)
and 120595(sdot) is the digamma functionThe maximum likelihood estimate (MLE) of 120596 is
obtained by solving numerically the nonlinear equations119880120572(120596) = 0 119880
120579(120596) = 0 119880
119886(120596) = 0 and 119880
119887(120596) = 0 We can
use iterative techniques such as the Newton-Raphson typealgorithm to obtain the estimate We employ the subroutineNLMixed in SAS [15]
For interval estimation of (120572 120579 119886 119887) and tests of hypothe-ses on these parameters we obtain the observed informationmatrix since the expected information matrix is very com-plicated and will require numerical integration The 4 times 4
observed information matrix J(120596) is
J (120596) = minus(
L120572120572
L120572120579
L120572119886
L120572119887
sdot L120579120579
L120579119886
L120579119887
sdot sdot L119886119886
L119886119887
sdot sdot sdot L119887119887
) (83)
whose elements are given in Appendix BUnder conditions that are fulfilled for parameters in the
interior of the parameter space but not on the boundarythe asymptotic distribution of radic119899( minus 120596) is 119873
4(0K(120596)minus1)
where K(120596) is the expected information matrix The normalapproximation is valid if K(120596) is replaced by J() that is theobserved informationmatrix evaluated at Themultivariatenormal 119873
4(0 J()minus1) distribution can be used to construct
approximate confidence intervals for the model parametersThe likelihood ratio (LR) statistic can be used for testing thegoodness of fit of the BGHN distribution and for comparingthis distribution with some of its special submodels
We can compute the maximum values of the unrestrictedand restricted log-likelihoods to construct LR statistics fortesting some submodels of the BGHN distribution For
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 11
example we may use the LR statistic to check if the fit usingthe BGHN distribution is statistically ldquosuperiorrdquo to a fit usingthe modified Weibull or exponentiated Weibull distributionfor a given data set In any case hypothesis testing of the type1198670 120596 = 120596
0versus 119867 120596 =120596
0can be performed using LR
statistics For example the test of1198670 119887 = 1 versus119867 119887 = 1 is
equivalent to compare the EGHN and BGHN distributionsIn this case the LR statistic becomes 119908 = 2119897(
120579 119886
119887) minus
119897(120579 119886 1) where 120579 119886 and119887 are theMLEs under119867 and
120579 and 119886 are the estimates under119867
0 We emphasize that first-
order asymptotic results for LR statistics can be misleadingfor small sample sizes Future research can be conducted toderive analytical or bootstrap Bartlett corrections
7 A BGHN Model for Survival Data withLong-Term Survivors
In population based cancer studies cure is said to occur whenthe mortality in the group of cancer patients returns to thesame level as that expected in the general population Thecure fraction is of interest to patients and also a useful mea-surewhen analyzing trends in cancer patient survivalModelsfor survival analysis typically assume that every subject inthe study population is susceptible to the event under studyand will eventually experience such event if the follow-up issufficiently long However there are situations when a frac-tion of individuals are not expected to experience the event ofinterest that is those individuals are cured or not susceptibleFor example researchers may be interested in analyzing therecurrence of a disease Many individuals may never expe-rience a recurrence therefore a cured fraction of the pop-ulation exists Cure rate models have been used to estimatethe cured fraction These models are survival models whichallow for a cured fraction of individualsThesemodels extendthe understanding of time-to-event data by allowing for theformulation of more accurate and informative conclusionsThese conclusions are otherwise unobtainable from an analy-sis which fails to account for a cured or insusceptible fractionof the population If a cured component is not present theanalysis reduces to standard approaches of survival analysisCure rate models have been used for modeling time-to-eventdata for various types of cancers including breast cancernon-Hodgkins lymphoma leukemia prostate cancer andmelanoma Perhaps themost popular type of cure ratemodelsis the mixture model introduced by Berkson and Gage [16]and Maller and Zhou [17] In this model the population isdivided into two sub-populations so that an individual eitheris cured with probability 119901 or has a proper survival function119878(119909)with probability 1minus119901This gives an improper populationsurvivor function 119878lowast(119909) in the form of the mixture namely
119878lowast
(119909) = 119901 + (1 minus 119901) 119878 (119909) 119878 (infin) = 0 119878lowast
(infin) = 119901
(84)
Common choices for 119878(119909) in (84) are the exponential andWeibull distributions Here we adopt the BGHN distribu-tion Mixture models involving these distributions have beenstudied by several authors including Farewell [18] Sy andTaylor [19] and Ortega et al [20] The book by Maller and
Zhou [17] provides a wide range of applications of the long-term survivor mixture model The use of survival modelswith a cure fraction has become more and more frequentbecause traditional survival analysis do not allow for model-ing data in which nonhomogeneous parts of the populationdo not represent the event of interest even after a long follow-up Now we propose an application of the BGHN distribu-tion to compose a mixture model for cure rate estimationConsider a sample 119909
1 119909
119899 where 119909
119894is either the observed
lifetime or censoring time for the 119894th individual Let a binaryrandom variable 119911
119894 for 119894 = 1 119899 indicating that the 119894th
individual in a population is at risk or not with respect toa certain type of failure that is 119911
119894= 1 indicates that the
119894th individual will eventually experience a failure event(uncured) and 119911
119894= 0 indicates that the individual will never
experience such event (cured)For an individual the proportion of uncured 1minus119901 can be
specified such that the conditional distribution of 119911 is given byPr(119911
119894= 1) = 1minus119901 Suppose that the119883
119894rsquos are independent and
identically distributed random variables having the BGHNdistribution with density function (4)
The maximum likelihood method is used to estimate theparametersThus the contribution of an individual that failedat 119909
119894to the likelihood function is given by
(1 minus 119901) 2119887minus1
radic2120587 (120572119909119894) (119909
119894120579)
120572 exp [minus (12) (119909119894120579)
2120572
]
119861 (119886 119887)
times 2Φ [(
119909119894
120579
)
120572
] minus 1
119886minus1
1 minus Φ[(
119909119894
120579
)
120572
]
119887minus1
(85)
and the contribution of an individual that is at risk at time 119909119894
is
119901 + (1 minus 119901) 1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887) (86)
The new model defined by (85) and (86) is called the BGHNmixture model with long-term survivors For 119886 = 119887 = 1 weobtain the GHN mixture model (a new model) with long-term survivors
Thus the log-likelihood function for the parameter vector120579 = (119901 120572 120579 119886 119887)
119879 follows from (85) and (86) as
119897 (120579) = 119903 log[(1 minus 119901) 2
119887minus1radic2120587
119861 (119886 119887)
] + sum
119894isin119865
log( 120572
119909119894
)
+ 120572sum
119894isin119865
(
119909119894
120579
) minus
1
2
sum
119894isin119865
log [(119909119894
120579
)
2120572
]
+ (119886 minus 1)sum
119894isin119865
log 2Φ[(
119909119894
120579
)
120572
] minus 1
+ (119887 minus 1)sum
119894isin119865
log 1 minus Φ[(
119909119894
120579
)
120572
]
+ sum
119894isin119862
log 119901 + (1 minus 119901) [1 minus 1198682Φ[(119909
119894120579)120572]minus1
(119886 119887)]
(87)
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Probability and Statistics
where119865 and119862denote the sets of individuals corresponding tolifetime observations and censoring times respectively and 119903is the number of uncensored observations (failures)
8 Applications
In this section we give four applications using well-knowndata sets (three with censoring and one uncensored data set)to demonstrate the flexibility and applicability of the pro-posed model The reason for choosing these data is that theyallow us to show how in different fields it is necessary to havepositively skewed distributions with nonnegative supportThese data sets present different degrees of skewness and kur-tosis
81 Censored Data Transistors The first data set consists ofthe lifetimes of 119899 = 34 transistors in an accelerated life testThree of the lifetimes are censored so the censoring fractionis 334 (or 88) Wilk et al [21] stated that ldquothere is reasonfrom past experience to expect that the gamma distributionmight reasonably approximate the failure time distributionrdquoWilk et al [21] and also Lawless [22] fitted a gammadistribution Alternatively we fit the BGHN model to thesedata We estimate the parameters 120572 120579 119886 and 119887 by maximumlikelihood The MLEs of 120572 and 120579 for the GHN distributionare taken as starting values for the iterative procedure Thecomputations were performed using the NLMixed procedurein SAS [15] Table 2 lists the MLEs (and the correspondingstandard errors in parentheses) of the model parameters andthe values of the following statistics for some models AkaikeInformationCriterion (AIC) Bayesian InformationCriterion(BIC) and Consistent Akaike Information Criterion (CAIC)The results indicate that the BGHNmodel has the lowest AICBIC and CAIC values and therefore it could be chosen asthe best model Further we calculate the maximum unre-stricted and restricted log-likelihoods and the LR statisticsfor testing some submodels For example the LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHNmodels is 119908 =
2minus1201 minus (minus1260) = 118 (119875 value =00027) which givesfavorable indication towards the BGHNmodel In special theWeibull density function is given by
119891 (119909) =
120574
120582120574119909120574minus1 exp [minus(119909
120582
)
120574
] 119909 120582 120574 gt 0 (88)
Figure 1(a) displays plots of the empirical survival functionand the estimated survival functions of the BGHN GHNandWeibull distributions We obtain a good fit of the BGHNmodel to these data
82 Censored Data Radiotherapy The data refer to thesurvival time (days) for cancer patients (119899 = 51) undergoingradiotherapy [23] The percentage of censored observationswas 1765 Fitting the BGHN distribution to these data weobtain the MLEs of the model parameters listed in Table 3The values of the AIC CAIC and BIC statistics are smallerfor the BGHN distribution as compared to those values ofthe GHN and Weibull distributions The LR statistic fortesting the hypotheses 119867
0 119886 = 119887 = 1 versus 119867
1 119867
0is not
true that is to compare the BGHN and GHN models is119908 = 2minus2933 minus (minus2994) = 122 (119875 value =00022) whichyields favorable indication towards the BGHN distributionThus this distribution seems to be a very competitive modelfor lifetime data analysis Figure 1(b) displays the plots ofthe empirical survival function and the estimated survivalfunctions for the BGHN GHN and Weibull distributionsWe note a good fit of the BGHN distribution
83 Uncensored Data USS Halfbeak Diesel Engine Herewe compare the fits of the BGHN GHN and Weibulldistributions to the data set presented byAscher and Feingold[24] from a USS Halfbeak (submarine) diesel engine Thedata denote 73 failure times (in hours) of unscheduledmaintenance actions for the USS Halfbeak number 4 mainpropulsion diesel engine over 25518 operating hours Table4 gives the MLEs of the model parameters The values ofthe AIC CAIC and BIC statistics are smaller for the BGHNdistribution when compared to those values of the GHNand Weibull distributions The LR statistic for testing thehypotheses119867
0 119886 = 119887 = 1 versus119867
1119867
0is not true that is to
compare the BGHN and GHN models is 119908 = 2minus1961 minus
(minus2172) = 421 (119875 value lt00001) which indicates thatthe first distribution is superior to the second in termsof model fitting Figure 1(c) displays plots of the empiricalsurvival function and the estimated survival function of theBGHN GHN and Weibull distributions In fact the BGHNdistribution provides a better fit to these data
84 Melanoma Data with Long-Term Survivors In this sec-tion we provide an application of the BGHN model tocancer recurrence The data are part of a study on cutaneousmelanoma (a type of malignant cancer) for the evaluationof postoperative treatment performance with a high doseof a certain drug (interferon alfa-2b) in order to preventrecurrence Patients were included in the study from 1991to 1995 and follow-up was conducted until 1998 The datawere collected by Ibrahim et al [25] The survival time 119879is defined as the time until the patientrsquos death The originalsample size was 119899 = 427 patients 10 of whom did not presenta value for explanatory variable tumor thickness When suchcases were removed a sample of size 119899 = 417 patients wasretained The percentage of censored observations was 56Table 5 lists the MLEs of the model parametersThe values ofthe AIC CAIC and BIC statistics are smaller for the BGHNmixture model when compared to those values of the GHNmixturemodelThe LR statistic for testing the hypotheses119867
0
119886 = 119887 = 1 versus 1198671 119867
0is not true that is to compare the
BGHN and GHNmixture models becomes119908 = 2minus52475minus
(minus53405) = 186 (119875 value lt00001) which indicates thatthe BGHN mixture model is superior to the GHN mixturemodel in terms of model fitting Figure 2 provides plots ofthe empirical survival function and the estimated survivalfunctions of the BGHN and GHN mixture models Notethat the cured proportion estimated by the BGHN mixturemodel (119901BGHN = 04871) is more appropriate than that oneestimated by the GHN mixture model (119901GHN = 05150)Further the BGHN mixture model provides a better fit tothese data
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 13
0 10 50403020
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
x
S(x)
(a)
0 140012001000800600400200
Kaplan-MeierGHN
BGHNWeibull
10
08
06
04
02
00
S(x)
x
(b)
0 252015105
10
08
06
04
02
00
S(x)
x
Kaplan-MeierGHN
BGHNWeibull
(c)
Figure 1 Estimated survival function for the BGHN GHN andWeibull distributions and the empirical survival (a) for transistors data (b)for radiotherapy data and (c) USS Halfbeak diesel engine data
Table 2 MLEs of the model parameters for the transistors data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00479 (00062) 193753 (28892) 40087 (04000) 19213 (02480) 2482 2496 2543GHN 09223 (01361) 257408 (38627) 1 (mdash) 1 (mdash) 2560 2564 2591
120582 120574
Weibull 207819 (28215) 13414 (01721) 2634 2678 2704
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Journal of Probability and Statistics
Table 3 MLEs of the model parameters for the radiotherapy data the corresponding SEs (given in parentheses) and the AIC CAIC andBIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 00456 (00051) 54040 (10819) 22366 (04313) 11238 (01560) 5946 5955 6023GHN 07074 (00879) 53345 (886917) 1 (mdash) 1 (mdash) 6028 6031 6067
120582 120574
Weibull 36176 (524678) 10239 (01062) 7057 7060 7096
Table 4 MLEs of the model parameters for the USS Halfbeak diesel engine data the corresponding SEs (given in parentheses) and the AICCAIC and BIC statistics
Model 120572 120579 119886 119887 AIC CAIC BICBGHN 70649 (00027) 189581 (00027) 02368 (00416) 01015 (00134) 4002 4016 4093GHN 38208 (04150) 218838 (04969) 1 (mdash) 1 (mdash) 4384 4390 4429
120582 120574
Weibull 211972 (06034) 42716 (04627) 4555 4561 4601
Table 5 MLEs of the model parameters for the melanoma data the corresponding SEs (given in parentheses) and the AIC CAIC and BICstatistics
Model 119901 120572 120579 119886 119887 AIC CAIC BICBGHNMixture 04871 (00349) 02579 (00197) 548258 (172126) 194970 (10753) 409300 (22931) 10595 10596 10796GHNMixture 05150 (00279) 12553 (00799) 25090 (01475) 1 (mdash) 1 (mdash) 10741 10742 10862
9 Concluding Remarks
In this paper we discuss somemathematical properties of thebeta generalized half normal (BGHN) distribution introducedrecently by Pescim et al [3] The incorporation of additionalparameters provides a very flexible model We derive apower series expansion for the BGHN quantile function Weprovide some new structural properties such as momentsgenerating function mean deviations Renyi entropy andreliability We investigate properties of the order statisticsThe method of maximum likelihood is used for estimatingthe model parameters for uncensored and censored data Wealso propose a BGHNmodel for survival data with long-termsurvivors Applications to four real data sets indicate that theBGHN model provides a flexible alternative to (and a betterfit than) some classical lifetimes models
Appendices
A Appendix A
Here we derive some properties of the BGHNorder statisticsUsing the identity in Gradshteyn and Ryzhik [5] and aftersome algebraic manipulations we obtain an alternative formfor the density function of the 119894th order statistic namely
119891119894119899(119909)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 [119886 (119894 + 119896) + 119895 119887] 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 1 + 119894)
times 119891119894119895119896
(119909)
(A1)
Kaplan-MeierGHN mixtureBGHN mixture
10
09
08
07
06
05
04
0 2 4 6 8
P = 04871
S(x)
x
Figure 2 Estimated survival functions for the BGHN and GHNmixture models and the empirical survival for the melanoma data
where 119891119894119895119896
(119909) denotes the density of the BGHN (120572 120579 119886(119894 +
119896) + 119895 119887) distribution and the quantity 119889119894119895119896
is defined byPescim et al [3]
From (A1) we obtain alternative expressions for themoments and mgf of the BGHN order statistics given by
119864 (119883119904
119894119899)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119864 (119883119904
119894119895119896)
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 15
119872119894119899(119905)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times119872120572120579119886119887
(119905)
(A2)
The mean deviations of the order statistics in relation tothe mean and to the median can be expressed as
1205751(119883
119894119899) = 2120583
119894119865119894119899(120583
119894) minus 2120583
119894+ 2119869
119894(120583
119894)
1205752(119883
119894119899) = 2119869
119894(119872
119894) minus 120583
119894
(A3)
respectively where 120583119894= 119864(119883
119894119899) 120583
119894= Median(119883
119894119899) and 119869
119894(119902)
are defined in Section 54 We use (A1) to obtain 119869119894(119902)
119869119894(119902)
=
119899minus119894
sum
119896=0
infin
sum
119895=0
(minus1)119896
(119899minus119894
119896) Γ(119887)
119894+119896minus1
119861 (119886 (119894 + 119896) + 119895 119887) 119889119894119895119896
119861(119886 119887)119894+119896
119861 (119894 119899 minus 119894 + 1)
times 119879 (119902)
(A4)
where 119879(119902) comes from (35)Bonferroni and Lorenz curves of the order statistic are
defined in Section 54 by
119861119894119899(120588) =
1
120588120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
119871119894119899(120588) =
1
120583119894
int
119902
0
119909119891119894119899(119909) 119889119909
(A5)
respectively where 119902 = 119865minus1
119894119899(120588) From int
119902
0
119909119891119894119899(119909)119889119909 = 120583
119894minus
119869119894(120588) these curves also can be expressed as
119861119894119899(120588) =
1
120588
minus
119869119894(120588)
120588120583119894
119871119894119899(120588) = 1 minus
119869119894(120588)
120583119894
(A6)
Using (A4) we obtain the Bonferroni and Lorenz curves forthe BGHN order statistics
B Appendix B
Here we give the necessary formulas to obtain the second-order partial derivatives of the log-likelihood function Aftersome algebraic manipulations we obtain
L120572120572
= 2sum
119894isin119865
(
119909119894
120579
)
2120572
log2 (119909119894
120579
) +
2 (119886 minus 1)
radic2120587
times
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894log (119909
119894120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus1
)times([1minus119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
)
times ([1 minus 119876 (119909119894)])
minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
]
times [119875(119909119894)]119886minus1
[119863 (119909119894)]
119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
[ (V119894log(
119909119894
120579
) [119875 (119909119894)](119886minus1)
times [119863 (119909119894)](119887minus1)
) times (1 minus 119876 (119909119894))
minus1
]
2
L120572120579= minus
119903
120579
+ 2 (
120572
120579
)sum
119894isin119865
(
119909119894
120579
)
2120572
log(119909119894
120579
) +
1
120579
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log2 (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Journal of Probability and Statistics
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579) [1 minus (119909
119894120579)
2120572
] + V119894120579
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894log2 (
119909119894
120579
) [1 minus (
119909119894
120579
)
2120572
] +
V119894
120579
times [119875 (119909119894)]119886minus1
[119863 (119909119894)]
119887minus1
)times(1minus119876 (119909119894))minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus2
times [119863 (119909119894)]119887minus1
)
times (1 minus 119876 (119909119894))minus1
+
(1 minus 119887)
radic120587
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus2
)
times ([1 minus 119876 (119909119894)])
minus1
+ 2119887minus1
sum
119894isin119862
(V2119894(
120572
120579
) log(119909119894
120579
) [119875 (119909119894)]2(119886minus1)
times [119863 (119909119894)]2(119887minus1)
)
times([1 minus 119876 (119909119894)]2
)
minus1
L120572119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894log (119909
119894120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119886V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120572119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894log (119909
119894120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120572
[1 minus 119876 (119909119894)]
minus 2119887minus1
sum
119894isin119862
( 119868119875(119909119894)(119886 119887) |
119887V119894log(
119909119894
120579
) [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)]2
)
minus1
L120579120579= 119903 (
120572
1205792) minus
120572
1205792sum
119894isin119865
(
119909119894
120579
)
2120572
minus 2(
120572
120579
)
2
sum
119894isin119865
(
119909119894
120579
)
2120572
+
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119875 (119909119894)
minus
2
radic2120587
sum
119894isin119865
[
V119894(120572120579)
119875 (119909119894)
]
2
+
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579) [(119909
119894120579)
2120572
minus 1120579 minus 1]
119863 (119909119894)
+
1
radic120587
sum
119894isin119865
[
V119894(120572120579)
119863 (119909119894)
]
2
minus 2119887minus1
sum
119894isin119862
(V119894(
120572
120579
) [(
119909119894
120579
)
2120572
minus
1
120579
minus 1] [119875 (119909119894)]119886minus1
times [119863 (119909119894)]119887minus1
) times ([1 minus 119876 (119909119894)])
minus1
+
2 (119886 minus 1)
radic2120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus2
[119863 (119909119894)]119887minus1
[1 minus 119876 (119909119894)]
+
(1 minus 119887)
radic120587
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]119886minus1
[119863 (119909119894)]119887minus2
[1 minus 119876 (119909119894)]
+2119887minus1
sum
119894isin119862
V2119894(120572120579)
2
[119875 (119909119894)]2(119886minus1)
[119863 (119909119894)]2(119887minus1)
[1 minus 119876 (119909119894)]2
L120579119886=
2 (119886 minus 1)
radic2120587
sum
119894isin119865
V119894(120572120579)
119875 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886120579
[1 minus 119876 (119909119894)]
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
L120579119887=
(1 minus 119887)
radic120587
sum
119894isin119865
V119894(120572120579)
119863 (119909119894)
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887120579
1 minus 119876 (119909119894)
2119887minus1
times sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887V119894(120572120579) [119875 (119909
119894)](119886minus1)
[119863 (119909119894)](119887minus1)
[1 minus 119876 (119909119894)]2
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 17
L119886119886= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119886)] minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119886
1 minus 119876 (119909119894)
]
2
L119886119887= 119903120595
1015840
(119886 + 119887) minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119886119887
1 minus 119876 (119909119894)
minus sum
119894isin119862
[ 119868119875(119909119894)(119886 119887) |
119886] [ 119868
119875(119909119894)(119886 119887) |
119887]
[1 minus 119876 (119909119894)]2
L119887119887= 119903 [120595
1015840
(119886 + 119887) minus 1205951015840
(119887)]
minus sum
119894isin119862
119868119875(119909119894)(119886 119887) |
119887
[1 minus 119876 (119909119894)]
minus sum
119894isin119862
[
119868119875(119909119894)(119886 119887) |
119887
1 minus 119876 (119909119894)
]
2
(B1)
where
119868119875(119909119894)(119886 119887) |
119886120572=
1205972
120597119886120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120572=
1205972
120597119887120597120572
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886120579=
1205972
120597119886120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119887120579=
1205972
120597119887120597120579
[119876 (119909119894)]
119868119875(119909119894)(119886 119887) |
119886=
1205972
1205971198862[119876 (119909
119894)]
119868119875(119909119894)(119886 119887) |
119887=
1205972
1205971198872[119876 (119909
119894)]
(B2)
Acknowledgments
The authors would like to thank the editor the associateeditor and the referees for carefully reading the paper andfor their comments which greatly improved the paper
References
[1] K Cooray and M M A Ananda ldquoA generalization of the half-normal distribution with applications to lifetime datardquo Com-munications in Statistics Theory and Methods vol 37 no 8-10pp 1323ndash1337 2008
[2] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics Theory andMethods vol 31 no 4 pp 497ndash512 2002
[3] R R Pescim C G B Demetrio G M Cordeiro E M MOrtega and M R Urbano ldquoThe beta generalized half-normal
distributionrdquo Computational Statistics amp Data Analysis vol 54no 4 pp 945ndash957 2010
[4] G Steinbrecher ldquoTaylor expansion for inverse error functionaround originrdquo University of Craiova working paper 2002
[5] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts ElsevierAcademic Press Amsterdam The Nether-lands 7th edition 2007
[6] P F Paranaıba E M M Ortega G M Cordeiro and R RPescim ldquoThe beta Burr XII distribution with application tolifetime datardquo Computational Statistics amp Data Analysis vol 55no 2 pp 1118ndash1136 2011
[7] S Nadarajah ldquoExplicit expressions for moments of order sta-tisticsrdquo Statistics amp Probability Letters vol 78 no 2 pp 196ndash205 2008
[8] J Kurths A Voss P Saparin A Witt H J Kleiner and NWessel ldquoQuantitative analysis of heart rate variabilityrdquo Chaosvol 5 no 1 pp 88ndash94 1995
[9] D Morales L Pardo and I Vajda ldquoSome new statistics for test-ing hypotheses in parametric modelsrdquo Journal of MultivariateAnalysis vol 62 no 1 pp 137ndash168 1997
[10] A Renyi ldquoOn measures of entropy and informationrdquo inProceedings of the 4th Berkeley Symposium on Math Statistand Prob Vol I pp 547ndash561 University of California PressBerkeley Calif USA 1961
[11] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics Wiley Series in Probability andMathematical Statistics Probability and Mathematical Statis-tics John Wiley amp Sons New York NY USA 1992
[12] H A David and H N Nagaraja Order Statistics Wiley Seriesin Probability and Statistics John Wiley amp Sons Hoboken NJUSA 3rd edition 2003
[13] M Ahsanullah and V B Nevzorov Order Statistics Examplesand Exercises Nova Science Hauppauge NY USA 2005
[14] R Development Core Team R A Language and EnvironmentFor Statistical Computing R Foundation For Statistical Comput-ing Vienna Austria 2013
[15] SAS Institute SASSTAT Userrsquos Guide Version 9 SAS InstituteCary NC USA 2004
[16] J Berkson and R P Gage ldquoSurvival curve for cancer patientsfollowing treatmentrdquo Journal of the American Statistical Associ-ation vol 88 pp 1412ndash1418 1952
[17] R A Maller and X Zhou Survival Analysis with Long-TermSurvivors Wiley Series in Probability and Statistics AppliedProbability and Statistics John Wiley amp Sons Chichester UK1996
[18] V T Farewell ldquoThe use of mixture models for the analysis ofsurvival data with long-term survivorsrdquo Biometrics vol 38 no4 pp 1041ndash1046 1982
[19] J P Sy and J M G Taylor ldquoEstimation in a Cox proportionalhazards cure modelrdquo Biometrics vol 56 no 1 pp 227ndash2362000
[20] E M M Ortega F B Rizzato and C G B DemetrioldquoThe generalized log-gamma mixture model with covariateslocal influence and residual analysisrdquo Statistical Methods ampApplications vol 18 no 3 pp 305ndash331 2009
[21] M B Wilk R Gnanadesikan and M J Huyett ldquoEstimationof parameters of the gamma distribution using order statisticsrdquoBiometrika vol 49 pp 525ndash545 1962
[22] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsHoboken NJ USA 2nd edition 2003
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Journal of Probability and Statistics
[23] F Louzada-Neto JMazucheli and J AAchcarUma Introducaoa Analise de Sobrevivencia e Confiabilidade vol 28 JornadasNacionales de Estadıstica Valparaıso Chile 2001
[24] H Ascher and H Feingold Repairable Systems Reliability vol7 of Lecture Notes in Statistics Marcel Dekker New York NYUSA 1984
[25] J G IbrahimM-HChen andD SinhaBayesian Survival Ana-lysis Springer Series in Statistics Springer New York NY USA2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
