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Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference 142 (2012) 601–607

0378-37

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/jspi

Short communication

Interval estimation of PðYoXÞ for generalized Pareto distribution

Augustine Wong

Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario, Canada M3J 1P3

a r t i c l e i n f o

Article history:

Received 10 January 2011

Received in revised form

13 April 2011

Accepted 20 April 2011Available online 24 August 2011

Keywords:

Canonical parameter

Exponential family model

Modified signed log-likelihood ratio

statistic

Reliability

Stress–strength model

58/$ - see front matter & 2011 Elsevier B.V. A

016/j.jspi.2011.04.024

ail address: august@yorku.ca

a b s t r a c t

This paper deals with the interval estimation of PðY oXÞ when X and Y are two

independent generalized Pareto random variables with a common scale parameters. An

asymptotic confidence interval for PðYoXÞ is constructed based on the modified signed

log-likelihood ratio statistic. Monte Carlo simulations are performed to illustrate the

accuracy of the proposed method.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The problem of obtaining an interval estimate of the stress–strength reliability R¼ PðYoXÞ is of particular interest inengineering statistics and in biostatistics. For example, in a reliability study, let Y be the strength of a system and X be thestress applied to the system. Then R measures the chance that the system fails. Alternatively, in a clinical study, let Y be theresponse of a control group and X be the response of a treatment group. Then R measures the effectiveness of the treatment.

Inference about R under various assumptions on X and Y has been examined frequently in the statistical literature.Rezaei et al. (2010) gave a detail summary of recent development in this area. In particular, Rezaei et al. (2010) assumed X

and Y be independent generalized Pareto random variables with a common scale parameter and different shapeparameters. They derived an asymptotic distribution of the maximum likelihood estimator (MLE) of R and, hence,asymptotic confidence interval of R is obtained.

In this paper, we consider the model examined by Rezaei et al. (2010). A modified signed log-likelihood ratio statisticbased method, rnðRÞ, is proposed to construct an asymptotic confidence interval for R. The usual asymptotic methods basedon MLE, R, and the signed log-likelihood ratio statistic, r(R), have accuracy Oðn�1=2Þ; that is

R�RffiffiffiffiffiffiffiffiffiffiffiffiffiffivarðRÞ

q �Nð0;1ÞþOðn�1=2Þ,

rðRÞ �Nð0;1ÞþOðn�1=2Þ:

On the other hand, the proposed method has accuracy Oðn�3=2Þ; that is

rnðRÞ �Nð0;1ÞþOðn�3=2Þ:

ll rights reserved.

A. Wong / Journal of Statistical Planning and Inference 142 (2012) 601–607602

This paper is organized as follows: In Section 2, the modified signed log-likelihood statistic based method is discussed.In Section 3, we demonstrate how the proposed method can be applied to obtain asymptotic confidence interval of R whenX and Y are two independent generalized Pareto random variables with a common scale parameter. Results of Monte Carlosimulation studies are recorded in Section 4 that indicate the accuracy of the proposed method. Some concluding remarksare given in Section 5.

2. Approximate interval estimators

Let w¼ ðw1, . . . ,wnÞ be a sample from a distribution with log-likelihood function ‘ðyÞ ¼ ‘ðy;wÞ, where y is a vectorparameter with dimðyÞ ¼ p. Also let c¼cðyÞ be a scalar parameter of interest. Denote y be the overall MLE of y, which isobtained by solving ‘yðyÞ ¼ ð@‘ðyÞ=@yÞ

��y ¼ y ¼ 0. Moreover, denote yc be the constrained MLE of y for a given cðyÞ ¼c. In

general, yc can be obtained by the Lagrange multiplier technique: Let

Hðy,tÞ ¼ ‘ðyÞþt½cðyÞ�c�:

Then yc and t satisfied the first order conditions:

Hyðyc,tÞ ¼ @Hðy,tÞ@y

����ðy,tÞ ¼ ðyc ,tÞ

¼ 0,

Htðyc,tÞ ¼ @Hðy,tÞ@t

����ðy,tÞ ¼ ðyc ,tÞ

¼ 0:

Moreover, the tilted log-likelihood function is defined as

~‘ðyÞ ¼ ‘ðyÞþ t½cðyÞ�c�: ð1Þ

This tilted likelihood has the property ~‘ðycÞ ¼ ‘ðycÞ and is an important quantity for the proposed method.Two widely used methods for obtaining asymptotic confidence interval of c are based on the MLE of y and the signed

log-likelihood ratio statistic. It is well known that ðy�yÞ0varðyÞðy�yÞ is asymptotically distributed as w2-distribution with p

degrees of freedom, and varðyÞ can be estimated by the inverse of either the expected Fisher information matrix or theobserved information matrix evaluated at y. In practice, the latter,

^varðyÞ ¼ j�1yy0 ðyÞ ¼ ½�‘yy0 ðyÞ�

�1 ¼ �@2‘ðyÞ@y@y0

� ��1

y ¼ y

is preferred because of the simplicity in calculation. By applying the Delta method, we have

^varðcÞ ¼ ^varðcðyÞÞ �c0yðyÞ ^varðyÞcyðyÞ, ð2Þ

where cyðyÞ ¼ ð@cðyÞ=@yÞ9y ¼ y . Hence ðc�cÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^varðcÞ

qis asymptotically distributed as N(0,1). Thus a ð1�gÞ100%

confidence interval of c based on MLE is

c�zg=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^varðcÞ

q,cþzg=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^varðcÞ

q� �, ð3Þ

where zg=2 is the ð1�g=2Þ100 percentile of N(0,1).Alternatively, with regularity conditions stated by Cox and Hinkley (1974), the signed log-likelihood ratio statistic

rðcÞ ¼ sgnðc�cÞf2½‘ðyÞ�‘ðycÞ�g1=2 ¼ sgnðc�cÞf2½‘ðyÞ�~‘ðycÞ�g1=2 ð4Þ

is asymptotically distributed as N(0,1). Therefore, a ð1�gÞ100% confidence interval of c based on the signed log-likelihoodratio statistic is

fc : 9rðcÞ9rzg=2g: ð5Þ

Note that both methods have rates of convergence Oðn�1=2Þ only. In practice, the MLE based interval is often preferredbecause of the simplicity in calculations. Doganaksoy and Schmee (1993), however, illustrated that the signed log-likelihoodratio statistic based interval has better coverage property than the MLE based interval in cases that they examined.

In recent years, various adjustments to the signed log-likelihood ratio statistic have been proposed to improve theaccuracy of the signed log-likelihood ratio statistic. Reid (1996) and Severeni (2000) gave detail overview of thisdevelopment. In this paper, we consider the modified signed log-likelihood ratio statistic, which was introduced byBarndorff-Nielsen (1986, 1991) and has the form:

rnðcÞ ¼ rðcÞþrðcÞ�1logQ ðcÞrðcÞ

� , ð6Þ

where rðcÞ is the signed log-likelihood ratio statistic as defined in (4), and Q ðcÞ is a statistic that based on the log-likelihood function and an ancillary statistic. Barndorff-Nielsen (1986, 1991) showed that rnðcÞ is asymptotically

A. Wong / Journal of Statistical Planning and Inference 142 (2012) 601–607 603

distributed as N(0,1) with rate of convergence Oðn�3=2Þ. Hence, a 100ð1�gÞ% confidence interval of c based on rnðcÞ is

fc : 9rnðcÞ9rzg=2g: ð7Þ

However, for a general model, ancillary statistic might not exist, and even if it exists, it might not be unique. Fraser andReid (1995) established that Q ðcÞ is a standardized maximum likelihood departure in the canonical parameter scale andshowed that the general formula for the statistic Q ðcÞ required four major ingredients.

1.

The ancillary direction V:

V ¼@kðw,yÞ@w

� ��1 @kðw,yÞ@y

� ������y ¼ y

, ð8Þ

where kðw,yÞ is a pivotal quantity.

2. The locally defined canonical parameter:

jðyÞ ¼ @‘ðyÞ@w

V : ð9Þ

3.

The recalibrated parameter of interest in jðyÞ scale:

wðyÞ ¼cyðycÞj�1y ðycÞjðyÞ, ð10Þ

where jyðyÞ ¼ @jðyÞ=@y. Note that wðyÞ�wðycÞ is a measure of the maximum likelihood departure of c�c in jðyÞ scale.

4. Variance of the maximum likelihood departure of c�c in jðyÞ scale:

^varðwðyÞ�wðycÞÞ ¼s2ðwÞ9~jðyy0 ÞðycÞ9

9jðyy0 ÞðyÞ9,ð11Þ

where

9jðyy0 ÞðyÞ9¼ 9jyy0 ðyÞ99jyðyÞ9�2

,

9~jðyy0 ÞðycÞ9¼ 9~jyy0 ðycÞ99jyðycÞ9�2

,

s2ðwÞ ¼c0yðycÞ~j�1

yy0 ðycÞcyðycÞ,

~jyy0 ðycÞ ¼�@2 ~‘ðyÞ@y@y0

����y ¼ yc

:

And finally the general formula for Q ðcÞ is

Q ðcÞ ¼ sgnðc�cÞ9wðyÞ�wðyc9Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^varðwðyÞ�wðycÞÞ

q : ð12Þ

Therefore, rnðcÞ can be obtained from (6), and the confidence interval of c based on rnðcÞ is available from (7).

3. Confidence interval of PðY oXÞ for generalized Pareto distribution

As defined by Rezaei et al. (2010), the generalized Pareto distribution has density function:

f ðx;a,lÞ ¼ alð1þlxÞ�ðaþ1Þ x40

and cumulative distribution function:

PðXrxÞ ¼ Fðx;a,lÞ ¼ 1�ð1þlxÞ�a,

where a40 and l40 are the shape and scale parameters respectively. Then the stress–strength reliability R¼ PðYoXÞ

when X and Y are two independent generalized Pareto distributions with common scale parameter is

R¼ PðYoXÞ ¼

Z 10

Z x

0alð1þlxÞ�ðaþ1Þblð1þlyÞ�ðbþ1Þ dy dx¼

baþb

: ð13Þ

Let ðx1, . . . ,xnÞ and ðy1, . . . ,ymÞ be observed samples from two independent generalized Pareto distributions withcommon scale parameter. The log-likelihood function is

‘ðyÞ ¼ n log aþm log bþðnþmÞlog l�ðaþ1ÞXn

i ¼ 1

logð1þlxiÞ�ðbþ1ÞXm

j ¼ 1

logð1þlyjÞ,

A. Wong / Journal of Statistical Planning and Inference 142 (2012) 601–607604

where y¼ ða,b,lÞ0. By differentiating ‘ðyÞ with respect to y once and twice, we have the score function:

‘yðyÞ ¼@‘ðyÞ@y¼

na�Pn

i ¼ 1 logð1þlxiÞ

mb�Pm

j ¼ 1 logð1þlyjÞ

nþml �ðaþ1Þ

Pni ¼ 1

xi

1þlxi�ðbþ1Þ

Pmj ¼ 1

yj

1þlyj

0BB@

1CCA

and the observed information matrix

jyy0 ðyÞ ¼ �‘yy0 ðyÞ ¼�@2‘ðyÞ@y@y0

¼

n

a20

Pni ¼ 1

xi

1þlxi

0m

b2

Pmj ¼ 1

yj

1þlyj

Pni ¼ 1

xi

1þlxi

Pmj ¼ 1

yj

1þlyj

nþm

l2�ðaþ1Þ

Pni ¼ 1

x2i

ð1þlxiÞ2�ðbþ1Þ

Pmj ¼ 1

y2j

ð1þlyjÞ2

0BBBBBBBB@

1CCCCCCCCA: ð14Þ

To obtain the overall MLE of y, y ¼ ða,b,lÞ0, we have to solve ‘yðyÞ ¼ 0 and we have

a ¼ nPni ¼ 1 logð1þ lxiÞ

ð15Þ

b ¼mPm

j ¼ 1 logð1þ lyjÞð16Þ

l ¼nþm

nPn

i ¼ 1

xi

1þ lxiPni ¼ 1 logð1þ lxiÞ

þ

mPm

j ¼ 1

yj

1þ lyjPmj ¼ 1 logð1þ lyjÞ

þPn

i ¼ 1

xi

1þ lxi

þPm

j ¼ 1

yj

1þ lyj

: ð17Þ

Note that (17) is of the form gðlÞ ¼ l. Hence l is a fixed point solution of (17). Therefore l can be obtained iteratively by

gðlhÞ ¼ lhþ1, h¼ 0;1,2, . . . ,

where l0 is an initial estimate of l. The iteration procedure should be stopped when 9lhþ1�lh9 is smaller than some pre-determined tolerance level. It is important to note that this iteration process will converge to l but the convergenceprocess can be extremely slow. On the other hand, the standard gradient dependent type numerical methods, such as theNewton–Raphson method, may not converge to l. Once l is obtained, a,b and jyy0 ðyÞ can be obtained by (15), (16) and (14)respectively. Thus, by (3), a ð1�gÞ100% confidence interval of R based on MLE is

R�zg=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi^varðRÞ

q,Rþzg=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi^varðRÞ

q� �, ð18Þ

where

R ¼b

aþ b,

^varðRÞ ¼ R0

y ^varðyÞRy,

^varðyÞ ¼ j�1yy0 ðyÞ,

Ry ¼@R

@a ,@R

@b,@R

@l

� �0����y ¼ y¼ �

b

ðaþ bÞ2,

aðaþ bÞ2

,0

!0:

Similarly, the constrained MLE of y for a given R can be obtained from the Lagrange multiplier technique as described inSection 2, and hence we have

aR ¼1�R

RbR, ð19Þ

bR ¼nþm

1�R

R

Pni ¼ 1 logð1þ lRxiÞþ

Pmj ¼ 1 logð1þ lRyjÞ

, ð20Þ

A. Wong / Journal of Statistical Planning and Inference 142 (2012) 601–607 605

lR ¼ ðnþmÞ

ðnþmÞ1�R

R

Pni ¼ 1

xi

1þ lRxi

þPm

j ¼ 1

yj

1þ lRyj

" #

1�R

R

Pni ¼ 1 logð1þ lRxiÞþ

Pmj ¼ 1 logð1þ lRyjÞ

þXn

i ¼ 1

xi

1þ lRxi

þXm

j ¼ 1

yj

1þ lRyj

8>>>><>>>>:

9>>>>=>>>>;

�1

, ð21Þ

t ¼ bR

R

n

ð1�RÞbR

�1

R

Xn

i ¼ 1

logð1þ lRxiÞ

" #: ð22Þ

Again (21) is of the form gðlRÞ ¼ lR and thus lR is a fixed point solution of (21). With lR, the constrained MLE of y for agiven R, yR ¼ ðaR,bR,lRÞ

0 can be obtained. Therefore, by (5), a ð1�gÞ100% confidence interval of R based on the signed log-likelihood ratio statistic is

fR : 9rðRÞ9ozg=2g, ð23Þ

where

rðRÞ ¼ sgnðR�RÞf2½‘ðyÞ�‘ðyRÞ�g1=2:

Furthermore, the tilted log-likelihood function, as defined in (1), is

~‘ðyÞ ¼ ‘ðyÞþ t baþb

�R

� �

and

~jyy0 ðyÞ ¼�@2 ~‘ðyÞ@y@y0

¼ jyy0 ðyÞþ

�t 2bðaþbÞ3

t a�bðaþbÞ3

0

t a�bðaþbÞ3

t 2aðaþbÞ3

0

0 0 0

0BBBBB@

1CCCCCA,

where jyy0 ðyÞ is given in (14).A pivotal quantity for this problem is

kðx1, . . . ,xn,y1, . . . ,ym,yÞ ¼ ðFðx1;a,lÞ, . . . ,Fðxn;a,lÞ,Fðy1;b,lÞ, . . . ,Fðym;b,lÞÞ0

and therefore the ancillary direction V can be obtained from (8):

V ¼

�1þ lx1

allogð1þ lx1Þ 0 �

x1

l^ ^ ^

�1þ lxn

allogð1þ lxnÞ 0 �

xn

l

0 �1þ ly1

bllogð1þ ly1Þ �

y1

l^ ^ ^

0 �1þ lym

bllogð1þ lymÞ �

ym

l

0BBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCA

¼ ðv1 v2 v3Þ,

where vi is the ith column of V. Thus from (9), the locally defined canonical parameter is

jðyÞ ¼

�Pn

i ¼ 1ðaþ1Þl

1þlxiv1i

�Pm

j ¼ 1ðbþ1Þl

1þlyjv2j

�Pn

i ¼ 1ðaþ1Þl

1þlxiv3i�

Pmj ¼ 1ðbþ1Þ

l1þlyj

v3j

0BBBBBBBB@

1CCCCCCCCA

,

A. Wong / Journal of Statistical Planning and Inference 142 (2012) 601–607606

where vst is the tth entry of vs, and

jyðyÞ ¼@jðyÞ@y¼

�Pn

i ¼ 1

l1þlxi

v1i 0 �Pn

i ¼ 1

aþ1

ð1þlxiÞ2v1i

0 �Pm

j ¼ 1

l1þlyj

v2j �Pm

j ¼ 1

bþ1

ð1þlyjÞ2v2j

�Pn

i ¼ 1

l1þlxi

v3i �Pm

j ¼ 1

l1þlyj

v3j �Pn

i ¼ 1

aþ1

ð1þlxiÞ2

v3i�Pm

j ¼ 1

bþ1

ð1þlyjÞ2v3j

0BBBBBBBBB@

1CCCCCCCCCA:

Furthermore wðyÞ can be obtained from (10), and the standardized maximum likelihood departure in jðyÞ scale can beobtained from (12). Finally, from (6), we have

rnðRÞ ¼ rðRÞþrðRÞ�1logQ ðRÞ

rðRÞ

� ð24Þ

and, from (7), a ð1�gÞ100% confidence interval of R based on the rnðRÞ is

fR : 9rnðRÞ9ozg=2g: ð25Þ

4. Numerical results

In this section, for a specific data set, we first obtained 95% confidence interval for R from the methods discussed in thispaper. Then Monte Carlo simulation studies are conducted to compare the accuracy of the methods.

Consider the model presented in Section 3 with the following data set:

Table

Comb

Com

1

2

3

4

5

TableThe 9

Met

(18)

(23)

(25)

2

ination of n,m

bination

15% confidence

hod 95% Co

(0.4826

(0.4594

(0.4323

,a,b, and l.

n

30

30

30

30

50

interval of R.

nfidence interv

, 0.9035)

, 0.8661)

, 0.8424)

m

30

30

50

50

40

al of R Leng

0.42

0.40

0.41

a

1.4

1

2

3

1.2

th

09

67

01

b

1

1.4

1.4

1.4

3

l

1

1

2

4

3

x

0.1730 1.2833 4.9105 0.2700 0.0822 1.3691 0.7097 2.2495 8.3866 0.0872

y

0.1731 0.4910 0.2113 0.5061 0.1645 0.1193 0.2778 1.9863 0.4989 1.5172

From solving (15)–(17), we have y ¼ ða,b,lÞ0 ¼ ð1:9276,4:3517,0:4902Þ0 and R ¼ 0:6930. Table 1 reported the 95%confidence interval of R obtained from the three methods discussed in this paper and the corresponding length of theconfidence intervals. Although (23) gives the shortest interval and (18) gives the longest interval, the length of the intervalis not our main interest as we are more interested in the coverage property of the confidence intervals. As we can see, the95% confidence interval of R is quite different. Therefore, it is important to study the coverage property of these methods.

Theoretically, as stated in Section 2, (18) and (23) have rates of convergence Oðn�1=2Þ and (25) has rate of convergenceOðn�3=2Þ. In other words, theoretically, (25) is a more accurate method than the other two methods. Numericalcomparisons are conducted via Monte Carlo simulation studies. We used 5000 simulated samples for the combinationsof n,m,a,b, and l listed in Table 2. For each simulated sample, a 95% confidence interval of R is calculated by each of themethod discussed in this paper. Table 3 recorded

�

Lower error: the proportion of the true R falling below the lower limit of the 95% confidence interval.Nominal level is 0.025 with standard error 0.0022. � Upper error: the proportion of the true R falling above the upper limit of the 95% confidence interval.

Nominal level is 0.025 with standard error 0.0022.

Table 3Results of simulation studies.

Combination Method Lower error Upper error Central coverage Average bias

1 (18) 0.0382 0.0316 0.9302 0.0099

(23) 0.0318 0.0298 0.9384 0.0058

(25) 0.0242 0.0266 0.9492 0.0012

2 (18) 0.0252 0.0432 0.9316 0.0092

(23) 0.0244 0.0322 0.9434 0.0039

(25) 0.0234 0.0266 0.9502 0.0015

3 (18) 0.0426 0.0214 0.9360 0.0106

(23) 0.0332 0.0218 0.9450 0.0057

(25) 0.0288 0.0216 0.9496 0.0036

4 (18) 0.0504 0.0154 0.9342 0.0175

(23) 0.0356 0.0224 0.9420 0.0066

(25) 0.0282 0.0246 0.9472 0.0018

5 (18) 0.0176 0.0476 0.9348 0.015

(23) 0.0244 0.0334 0.9422 0.0045

(25) 0.0264 0.0262 0.9474 0.0013

A. Wong / Journal of Statistical Planning and Inference 142 (2012) 601–607 607

�

Central coverage: the proportion of the true R falling within the 95% confidence interval.Nominal level is 0.95 with standard error 0.0030. � Average bias: average bias ¼ 9Lower error�0:0259þ9Upper error�0:0259=2

Nominal level is 0.

From Table 3, the modified signed log-likelihood ratio statistic based method (25) gives the best results—all within twostandard errors of the nominal values. The signed log-likelihood ratio statistic based method (23) gives good centralcoverage, but it gives asymmetric lower and upper errors. The MLE based method (18) does not give satisfactory results.

As a final note, Monte Carlo simulation studies on other combinations of n,m,a,b, and l have also been performed.Results are similar to those reported in Table 3 and are available from the author upon request. We also observed that thesimulation studies that results of the proposed method are not affected by the choice of l. As a word of warning, we haveto be very careful with the numerical methods for obtaining y and yR. Inaccurate estimation of y and yR might give non-positive definite or even singular jyy0 ðyÞ and ~jyy0 ðyRÞ, and thus the proposed method cannot be applied.

5. Conclusion

An asymptotic confidence interval for PðYoXÞ when X and Y are two independent generalized Pareto random variableswith a common scale parameter is proposed. The proposed method is based on the modified signed log-likelihood ratiostatistic. Simulation results illustrated the extreme accuracy of the proposed method. The proposed methodology can beextended to the non-equal scale parameters case. The numerical calculations will be more tedious as we have to work witha four dimensional parameter space. Finally, as long as the log-likelihood function and the pivotal quantity are availableexplicitly, the methodology can be applied to obtain inference for the stress–strength reliability.

References

Barndorff-Nielsen, O.E., 1986. Inference on full and partial parameters, based on the standardized signed log-likelihood ratio. Biometrika 73, 307–322.Barndorff-Nielsen, O.E., 1991. Modified signed log-likelihood ratio. Biometrika 78, 557–563.Cox, D.R., Hinkley, D.V., 1974. Theoretical Statistics. Chapman and Hall, London.Doganaksoy, N., Schmee, J., 1993. Comparisons of approximate confidence intervals for distributions used in life-data analysis. Technometrics 35,

175–184.Fraser, D.A.S., Reid, N., 1995. Ancillaries and third order significance. Utilitas Mathematica 7, 33–55.Reid, N., 1996. Likelihood and higher-order approximations to tail areas: a review and annotated bibliography. Canadian Journal of Statistics 24, 141–166.Rezaei, S., Tahmasbi, R., Mahmoodi, M., 2010. Estimation of PðY oXÞ for generalized Pareto distribution. Journal of Statistical Planning and Inference 140,

480–494.Severeni, T., 2000. Likelihood Methods in Statistics. Oxford University Press, New York.