International Journal of Risk Theorycefair/files/no9.pdf · proaches such as Bootstrap. Further, in 2010 they developed the exact likelihood for the same distribution but under joint

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International Journal of Risk Theory

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Alexandru Myller

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International Journal of Risk Theory

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Content

Mathematics and Informatics for Risk Theory

I. Băncescu1, S. Dedu, M. Stativa

Generalized Lindley populations under joint type II censored scheme

1

V. Preda, I. Băncescu

A new family of distributions with a general generic distribution for reliability

studies. Properties and Application

13

Author Guidelines 39

International Journal of Risk Theory, Vol 6 (no.1), 2016 1

Generalized Lindley populations under joint type II censoredscheme

I. Bancescu1∗, S. Dedu2, M. Stativa3

1 Doctoral School of Mathematics, University of Bucharest2 Bucharest University of Economic Studies, Bucharest

3 Faculty of Mathematics and Computer Science, University of Bucharest∗ Corresponding author: irina [email protected]

Abstract

This paper deals with the Bayesian and Non-Bayesian estimators in the case of two Lindley gen-eralized populations under joint type II censored scheme. We obtain the maximum likelihood estima-tors (MLEs) of the parameters and the corresponding asymptotic distributions. Bootstrap confidenceintervals have been constructed and Bayesian analysis for the parameters has be discussed. Theproblem of estimating reliability R = P(Y < X) in the case of the generalized Lindley distributionshas been considered. The asymptotic distribution of R has been used to construct an asymptoticconfidence interval. We have also proposed two bootstrap confidence intervals.

Keywords:generalized Lindley distribution, Bayesian analysis, joint type II censored scheme, stree-strenght model,maximum likelihood estimators, asymptotic distribution, bootstrap confidence intervals

1 Introduction

Recently various quantitative techniques for solving optimization problems which arise in economy,social sciences, engineering and many other domains have been developed. We can mention the con-tributions from [9, 19, 25, 26, 37]. The Lindley distribution was introduced by Lindley (1958) [23, 24]as a new distribution useful to analyze lifetime data especially in applications modeling stress-strengthreliability. Ghitany et al. (2008) [18] have discussed various properties of this distribution and showedthat in many ways that it provides a better model for some applications than the exponential distribution.They also showed in a numerical example that the Lindley distribution gives better modeling for waitingtimes and survival times data than the exponential distribution.

Recently, in [12] a generalized Lindley distribution GL(α, θ), α, θ > 0 was presented with the fol-lowing distribution function

F(x) = 1 − (1 +θx

αθ + 1)e−θx

and its density function


f (x) = e−θxθ2(α + x)αθ + 1

, x > 0, α, θ > 0 (1)

A joint censoring scheme can be used in life comparative tests of products from different units withinthe same factory. They are of great importance when it comes to ascerting their reliability. Due to thecost and economical time savings, the experiment can be terminated as soon as r fixed number failuresoccur.

Following, for example [6, 7] a joint censoring type II scheme is defined as follows. Suppose thatX1, ...Xm are the lifetimes of m specimens of product A and are independent identically distributedrandom variables from the distribution function F(x) and the density function f (x), and Y1, ...,Yn thelifetimes of n specimens of another product B, are independent identically distributed random variablesfrom distribution function G(x) and density function g(x).

We also suppose that W(1) < W(2) < ... < W(N) are the order statistics of the N = n + m randomvariables [X1, ...,Xm; Y1, ...,Yn]. Then under the joint Type II censoring scheme, the observable dataconsist of (Z,W), where W = (W(1),W(2), ...,W(r)), with r (1 ≤ r < N) being a pre-fixed integer, andZ = (Z1, ...,Zr) with zi = 1 or 0 according as wi is from an X− or Y− failure.

We suppose that the products which we want to test are manufactured by two different productionlines under the same conditions, and we select two independent samples of sizes m and n from thoselines, and the selected samples are placed at the same time on a life testing experiment. We will endthe experiment when a certain number of failures will occur. The successive failure times and the corre-sponding product types will be recorded, and the experiment will end as soon as a pre-fixed number offailures occur.

The likelihood inference under joint type censoring in the case of the exponential distribution hasbeen developed in 2008 by Balakrishnan and Rasouli [33]. They derived some methods which are basedon maximum likelihood estimates(MLE) and compared the performance with ones based on other ap-proaches such as Bootstrap. Further, in 2010 they developed the exact likelihood for the same distributionbut under joint progressive type-II censoring. Shafay et al. [35] in the year 2013 derived the Bayesianinference for the unknown parameters of two exponential populations under joint type II censoring theydeveloped with the use of squared error, linear exponential and general entropy loss functions. Finally in2010, Rasouli and Balakrishnan [33] work by considering a joint type II censored sample arising fromh independent exponential populations. Related problems have been widely discussed in the statisticalliterature.

Let Mr =

r∑i=1

Zi denote the number of X−failures in W and Nr =

r∑i=1

(1 − Zi) = r −Mr will be the

number of Y−failures in W, and the likelihood of (Z,W) is given in the year 2008 by Balakrishnan andRasouli [33] as it follows.

L =m!n!

(m −mr)!(n − nr)!

[ r∏i=1

( f (wi))zi(g(wi))1−zi

](F(wr))m−mr(G(wr))n−nr (2)

where F = 1 − F, G = 1 − G are the survival functions of the two populations.The rest of the paper is organized has follows. In Section 2 we discuss the maximum likelihood

estimators and its asymptotic distribution. Bootstrap confidence intervals are constructed in Section 3.Bayes estimators are derived in Section 4. The estimation of the reliability R = P(Y < X) with the use


of the asymptotic distribution to construct confidence intervals is presented in Section 5. Also, bootstrapintervals are derived. Finally, we end this paper with conclusions.

2 Maximum Likelihood Estimators

Let us suppose that the two populations are generalized Lindley distributed with the following distribu-tion functions

F1(x) = F(x;α1, θ1) = 1 −(1 +

θ1xα1θ1 + 1

)e−θ1x

and

F2(x) = G(x;α2, θ2) = 1 −(1 +

θ2xα2θ2 + 1

)e−θ2x

Its corresponding probability densities are

f (x) = f1(x) = e−θ1xθ21(α1 + x)

α1θ1 + 1,

g(x) = f2(x) = e−θ2xθ22(α2 + x)

α2θ2 + 1, αi, θi > 0, x > 0, i = 1, 2

In this case the likelihood function (2) becomes

L(α1, α2, θ1, θ2,w, z) =m!n!

(m −mr)!(n − nr)!

r∏i=1

{e−θ1wizi

θ2zi1 (α1 + wi)zi

(α1θ1 + 1)zi

× e−θ2wi(1 − zi)θ2(1−zi)

2 (α2 + wi)1−zi

(α2θ2 + 1)1−zi

}(1 +

θ1wr

α1θ1 + 1

)m−mr

× e−θ1wr(m −mr)(1 +

θ2wr

α2θ2 + 1

)n−nre−θ2wr(n − nr)

L(α1, α2, θ1, θ2,w, z) =m!n!θ2mr

1 θ2nr2

(m −mr)!(n − nr)!(α1θ1 + 1)mr(α2θ2 + 1)nr(3)

r∏i=1

{[e−θ1wi(α1 + wi)]zi[e−θ2wi(α2 + wi)]1−zi

}(1 +

θ1wr

α1θ1 + 1

)m−mr

× e−θ1wr(m −mr)(1 +

θ2wr

α2θ2 + 1

)n−nre−θ2wr(n − nr) (4)

In order to obtain the MLE’s of αi and θi we determine the first derivates of the natural logarithm ofthe likelihood function (4) with respect to αi and θi, i = 1, 2, and equating them to 0. So, we have thefollowing equations


lnL =k + 2mrlnθ1 + 2nrlnθ2 −mrln(α1θ1 + 1) − nrln(α2θ2 + 1) +

r∑i=1

ziln(α1 + wi)

− θ1

r∑i=1

ziwi +

r∑i=1

(1 − zi)ln(α2 + wi) − θ2

r∑i=1

wi(1 − zi)

+ (m −mr)ln(α1θ1 + θ1wr + 1) − (m −mr)ln(α1θ1 + 1) − θ1wr(m −mr)+ (n − nr)ln(α2θ2 + θ2wr + 1) − (n − nr)ln(α2θ2 + 1) − θ2wr(n − nr)

∂lnL∂α1

= −mrθ1

α1θ1 + 1+

r∑i=1

zi1

α1 + wi+ (m −mr)

[ θ1

α1θ1 + θ1wr + 1−

θ1

α1θ1 + 1

]= 0

∂lnL∂α2

= −nrθ2

α2θ2 + 1+

r∑i=1

1 − zi

α2 + wi+ (n − nr)

[ θ2

α2θ2 + θ2wr + 1−

θ2

α2θ2 + 1

]= 0

∂lnL∂θ1

=2mr

θ1−

mrα1

α1θ1 + 1−

r∑i=1

ziwi +(m −mr)(α1 + wr)α1θ1 + θ1wr + 1

−(m −mr)α1

α1θ1 + 1− wr(m −mr) = 0

∂lnL∂θ2

=2nr

θ2−

nrα2

α2θ2 + 1−

r∑i=1

(1 − zi)wi +(n − nr)(α2 + wr)α2θ2 + θ2wr + 1

−(n − nr)α2

α2θ2 + 1− wr(n − nr) = 0 (5)

The MLE’s of the parameters α1, α2, θ1 and θ2 can be obtained by solving the system of equations(5). Since the solution is not explicitly we can use the Newton-Raphson method.

We can derive the approximate asymptotic variance-covariance matrix for α1, α2, θ1 and θ2 byinverting the information matrix with the elements that are negative of the expected values at the secondorder derivates of logarithms of the likelihood functions. The Fisher information matrix associated I(λ) =I(α1, α2, θ1, θ2) is defined as follows

I(λ) = E

−∂2lnL∂α2

10 ∂2lnL

∂α1∂θ10

0 −∂2lnL∂α2

20 ∂2lnL

∂α2∂θ2

∂2lnL∂α1∂θ1

0 −∂2lnL∂θ2

10

0 ∂2lnL∂α2∂θ2

0 −∂2lnL∂θ2

2


where

∂2lnL∂α2

1

=mrθ2

1

(α1θ1 + 1)2 −

r∑i=1

zi

(α1 + wi)2 − (m −mr)θ1(θ1 + wr)

(α1θ1 + θ1wr + 1)2

+ (m −mr)θ2

1

(α1θ1 + 1)2

∂2lnL∂α2

2

=nrθ2

2

(α2θ2 + 1)2 −

r∑i=1

1 − zi

(α2 + wi)2 − (n − nr)θ2(θ2 + wr)

(α2θ2 + θ2wr + 1)2

+ (n − nr)θ2

2

(α2θ2 + 1)2

∂2lnL∂θ2

1

= −2mr

θ21

+mrα2

1

(α1θ1 + 1)2 −(m −mr)(α1 + wr)2

(α1θ1 + θ1wr + 1)2 +(m −mr)α2

1

(α1θ1 + 1)2

∂2lnL∂θ2

2

= −2nr

θ22

+nrα2

2

(α2θ2 + 1)2 −(n − nr)(α2 + wr)2

(α2θ2 + θ2wr + 1)2 +(n − nr)α2

2

(α2θ2 + 1)2

∂2lnL∂θ1∂α1

= −m

(α1θ1 + 1)2 +m −mr

(α1θ1 + θ1wr + 1)2

∂2lnL∂θ2∂α2

= −n

(α2θ2 + 1)2 +n − nr

(α2θ2 + θ2wr + 1)2

∂2lnL∂θ2∂α1

=∂2lnL∂θ1∂α2

= 0

The approximate 100(1−α)% confidence intervals for the parameters using the asymptotic normalityof the MLE’s can be obtained.

We suppose that λ is the MLE’s of the parameter vector λ = (α1, α2, θ1, θ2) = (λ1, λ2, λ3, λ4).Denote I0 = lim

n→∞nI(λ)−1.

Then λ is asymptotically normal distributed i.e√

n(λ − λ) ∼ N(0, I0)).Let s2

i = n−1ai j where ai, j are the elements in the matrix An = nI(λ)−1 and ˆI(λ) is the estimator forI(λ).

In this case, the asymptotic normality confidence intervals of λi, i = 1, 4 with the confidence level100(1 − α)% are given by

[λi ± z1− α2 si]

where z1− α2 denotes the upper 1 − α2 percentage point of the standard normal distribution.

3 Bootstrap Confidence Intervals

Now, using [16, 17] we will use some bootstrap methods which will help us to construct in our case forgeneralized Lindley distribution confidence intervals for the parameters λ1, λ2, λ3, λ4.

Bootstrap Percentile Interval Procedure (Boot-p)


This bootstrap method defines the lower and upper bounds of the confidence intervals using just the100α

2 th and the 100(1 − α2 )th quantiles of the empirical bootstrap distribution of λi.

Step 1 Compute the MLE λ = (λ1, ..., λ4) based on joint type II censored sample (w, z)Step 2 Use λi to generate a bootstrap joint type II censored sample (w∗, z∗) and compute the bootstrap

estimate of λi, say λ∗i , based on this bootstrap sample

Step 3 Repeat step 2, r0 times to have λ∗(1)i , λ∗(2)

i , ..., λ∗(r0)i

Step 4 Arrange λ∗(1)i , λ∗(2)

i , ..., λ∗(r0)i in ascending order to obtain λ∗[1]

i , λ∗[2]i , ..., λ∗[r0]

i

Step 5 A two sided 100(1−α)% percentile bootstrap confidence interval for λi, say[λ∗iL, λ

∗

iU

]is given

by

[λ∗iL, λ∗1iU)] =

[λ∗([r0α/2])

i , λ∗([r0(1− α2 )])i

]Studentized-t Interval Procedure (Boot-t)

The Boot-t confidence intervals estimators are computed according to the following steps:

Step 1 and Step 2 are same as the steps 1)-2) in Boot-pStep 3 Compute the t-statistic Tλ∗i

= (λ∗i − λi)/sλ∗i, where sλ∗i

is the bootstrap version

Step 4 Repeat steps 2)-3) r0 times and obtain T(1)λ∗i,T(2)

λ∗i, ...,T(r0)

λ∗i

Step 5 Arrange T(1)λ∗i,T(2)

λ∗i, ...,T(r0)

λ∗iin ascending order (statistic order) and obtain T[1]

λ∗i,T[2]

λ∗i, ...,T[r0]

λ∗i

Step 6 A two-sided 100(1 − α)% bootstrap-t confidence interval for λi, say[λ∗i,tL, λ

∗

i,tU

]is given by[

λi + T([r0α2 ])

λ∗isλi, λi + T([r0(1− α2 )])

λ∗isλi

], i = 1, 2

4 Bayes Estimators

Appling the Bayesian method we obtain the Bayesian estimators for the unknown parameters λi, i = 1, 4using symmetric squared error loss function and asymmetric LINEX loss functions.

Suppose that λi, i = 1, 4 have the following gamma prior distribution

π(λk) =bak

k

Γ(ak)λak−1

k exp(−bkλk), k = 1, 4, ak, bk > 0, λk > 0 (6)

From the Bayes theorem, (6) and (4) the joint posterior density is

l(data|λ) = C · L(λ1, λ2, λ3, λ4; w, z)4∏

k=1

π(λk)

where C is a normalization constant.


So, the Bayes estimator of any function of λi, i = 1, 4, say ϕ(λ1, λ2, λ3, λ4), under the squared errorloss function is ϕsqerr = Eλ1,λ2,λ3,λ4|data(ϕ(λ1, λ2, λ3, λ4)) and this is given by

ϕsqerr = C∫∞

0

∫∞

0

∫∞

0

∫∞

0ϕ(λ1, λ2, λ3, λ4)L(λ1, λ2, λ3, λ4,w, z)π(λk)dλk (7)

Under a Linex loss function, the Bayes estimate of a function ϕ(λ1, λ2, λ3, λ4) is given by

ϕlinex = −1c

lnE(e−cϕ(λ1,λ2,λ3,λ4)), c , 0 (8)

where

E(e−cϕ(λ1,λ2,λ3,λ4)) =

= c∫∞

0

∫∞

0

∫∞

0

∫∞

0e−cϕ(λ1,λ2,λ3,λ4)L(λ1, λ2, λ3, λ4,w, z)

4∏k=1

π(λk)dλk

The easiest method to solve equations (5), (7) and (8) is to solve them numerically.

5 Estimation of the stress-strength reliability R = P(Y < X)

The problem of estimating the reliability R = P(Y < X) in the stress-strength setting is very commonin the statistical literature. In the context of mechanical reliability R can be viewed as a measure of asystem performance when X is the strength of the system which is under a stress Y. The system fails ifthe applied stress is greater than its strength, if Y > X.

The estimation of R is widely extented in among statisticians. Owen, Cressewell and Hanson haveconsider this problem for the s-normally distributed, Awad et. al [3] have discussed the estimation forthe bivariate exponential distribution. More recently, for the bivariate distribution has been . Kundu andGupta (2005) [21] have analysied the problem when X and Y are two independent exponential, whileKundu and Raqab [34] have discussed the estimation of R with X and Y beeing independent identicallydistributed exponential, gamma or Burr type X. In 2013, Shun and Chien-Tai [8] considered the problemof estimating R = P(Y < X) with X and Y two independent but not identically distributed. In 2014,Ashour and Abo-Kasem [2] have discussed the estimation in the case of the two generalized exponentialdistribution.

We consider the problem of estimating reliability in the stress-strength model when the strengthof a unit or a system, X, has cumulative distribution function F1(x) and the stress subject to it, Y, hascumulative distribution function F2(y). So, X and Y are generalized Lindley random variables, but withdifferent parameters α1, θ1, and α2, θ2 respectively. The main aim of this section is to consider X and Yunder joint type II censoring scheme and to concentrate on the inference of R = P(Y < X). Confidenceintervals using the asymptotic distribution of R and two bootstrap resampling methods are explored aswell as MLE estimation.


5.1 Maximum Likelihood Estimation of Reliability R

We suppose that X and Y have generalized Lindley distributions with pdfs f1(x) and f2(y). X and Y haveF1(x) and F2(y) as their cumulative distribution functions in equation (2).

Now, we will consider the problem of estimating the reliability R = P(Y < X) based on a joint typeII censored sample. The reliability function is defined as

R = P(Y < X) =

∫∞

0F2(y) f1(x)dx

=θ2

1[(θ1 + θ2)2(α2θ2 + 1)(α1 − 1) − θ2(α1θ1 + 1) − θ22α1]

(α2θ2 + 1)(α1θ1 + 1)(θ1 + θ2)3

By using the MLE’s α1, α2, θ1, θ2 from the equations (5), we can obtain the MLE of R.

5.2 Confidence intervals for R

In this part we will show some methods for constructing confidence intervals for R.The asymptotic variance of R is

Var(R) = Q′

I(λ)−1Q

where Q′

= ( ∂R∂α1, ∂R∂α2, ∂R∂θ1, ∂R∂θ2

), Q′

denotes the transpuse of Q and the I(λ) Fisher information matrixpresented in equation (6), and

∂R∂α1

=θ2

1

[(θ1 + θ2)2(α2θ2 + 1)(θ1 + 1) − θ2

2

](α1θ1 + 1)2(α2θ2 + 1)(θ1 + θ2)3

∂R∂α2

=θ2

1θ22

[α1(θ1 + θ2) + 1

](α2θ2 + 1)2(α1θ1 + 1)(θ1 + θ2)3

∂R∂θ1

=θ1

(α2θ2 + 1)(α1θ1 + 1)2(θ1 + θ2)4

{(θ1 + θ2)2(α2θ1 + 1)(α1θ1 + 1)(α1 − 1)(θ1 + 2θ2)

− θ2(α1θ1 + 1)(θ1 + θ2)[θ1(α1 + θ1) + 2] − 3θ1θ2(α1θ1 + 1)[α1(θ1 + θ2) + 1]

− θ21θ

22α1(θ1 + θ2) − θ2

1(θ1 + θ2)(α2θ2 + 1)(α1 − 1)3}

∂R∂θ1

=θ2

1

(α2θ2 + 1)2(θ1 + θ2)4

{(θ1 + θ2)2(α2θ2 + 1)(α1 − 1)[2 − 3(α2θ2 + 1)]

+ α1(α2θ2 + 1)(θ1 + θ2)(θ1 + 2θ2) + (θ1 + θ2)[θ2(α1θ1 + 1)(α2 + 3) + θ22α1α2]

+ 3θ22α1(α2θ2 + 1)

}


If we replace its variance by its estimate, then we can obtain an approximate 100(1−α)% confidenceinterval for R as [

R ± z(1− α2 )

√Var(R)

]As in Section 3, using [16, 17], we consider the bootstrap confidence intervals for R.Parametric Boot-p confidence interval

In order to obtain the Boot-p confidence interval for R, we use the following algorithms:

Step 1 Based on the joint type II censored sample (w, z), compute the MLE’s λi, i = 1, 4 and then theMLE R for R

Step 2 Generate random samples from two independent generalized Lindley with parameters (λ1, λ2)and (λ3, λ4) of sizes m and n.

Then generate a bootstrap joint joint type II censored sample (w∗, z∗)Step 3 Compute the MLE R based on (w∗, z∗)Step 4 Repeat steps 2)-3) r0 times and obtain R∗(1), R∗(2), ..., R∗(r0)

Step 5 Arrange R∗(1), R∗(2), ..., R∗(r0) in ascending order to obtain the bootstrap sample (R∗[1], R∗[2], ..., R∗[r0])After that, a two sided 100(1 − α)% Boot-p confidence interval for R is given by[

R∗[r0α/2], R∗[r0(1− α2 )]]

Parametric Boot-t confidence interval

Step 1 Repeat steps 1)-3) from Parametric Boot-pStep 2 Compute Var(R∗) from equation (11) and compute the t-statistic

T∗ =R∗ − R√Var(R∗)

Step 3 Arrange T∗(1),T∗(2), ...,T∗(r0) in ascending order to obtain the bootstrap sample (T∗[1],T∗[2], ...,T∗[r0])Then a two-sided 100(1 − α)% Boot-t confidence interval for R is given by[

R + T∗[r0α/2]√

Var(R), R + T∗[r0(1− α2 )]√

Var(R)]

6 Conclusions

In this paper we have presented, discussed and constructed based on the generalized Lindley distributionand under the joint type II censoring scheme various statistical tools. First, we have obtain the maximumlikelihood estimators and its asymptotic distribution. Then we have constructed several bootstrap confi-dence intervals and derived the Bayesian estimators. Finally, we estimated the reliability R and proposedtwo confidence intervals for it. This results can be used in practice. The problems presented in this paperare of great use and very common in the literature.


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A new family of distributions with a general generic distributionfor reliability studies. Properties and Application

Vasile Preda1∗, Irina Bancescu2

1Department of Mathematics, University of Bucharest, Bucharest, Romania2Doctoral School of Mathematics, University of Bucharest, Bucharest, Romania

∗Corresponding author: [email protected]

AbstractWe study the mathematical properties of a new method of constructing probability distributions

based on the exponentiated generalized method and on the transmutation map. We derive the mo-ments, generatin function, Lorenz curve and other properties. We also derive the quantile function,the entropies and order statistics. We obtain new theorems concerning the limiting distributions ofthe order statistics and we show that they are not dependent of all parameters. We discuss maximumlikelihood estimation and stochastic order. We also perform data analysis for a new statistical model.Also we consider some new submodels of this class.

1 Introduction

The construction of new probability distributions suitable to model lifetime data and for many otherapplications is a important part in statistics. Along the years, statisticians have proposed many differentmethods of generalization of standard distributions, starting with a baseline distribution.

Recently, in 2013, Cordeiro et al.[10] have introduced a new method of adding two parameters to abaseline distribution. This method, namely the exponentiated generalized class of distributions, general-izes the exponentiated method introduced by Gupta et al. in 1999 and extends the idea first presented byLehmann in 1953.

The exponentiated generalized class of distributions (EG) is defined by

F(x) =[1 − {1 − G(x)}α

]βwhere α > 0, β > 0 are two additional shape parameters and G(x)

is a continuous cumulative distribution function (cdf). The corresponding density function of F(x) is

f (x) = αβg(x){1 − G(x)}α−1[1 − {1 − G(x)}α

]β−1.

Another method of constructing new probabibility distributions that has been in the attention ofstatisticians in the past years is the transmutation map developed by Shaw and Buckley (2007). Thetransmutation map using a continuous underlying distribution G(x) is defined as F(x) = (1 + λ)G(x) −λG(x)2, where |λ| ≤ 1 is the transmutation parameter. The corresponding density function is f (x) =g(x)[1 + λ − 2λG(x)].

In this paper, we propose a new method of constructing new probability distributions based on thetransmutation map and the exponentiated generalized class of distributions. This general approach gen-eralizes the exponentiated, the transmutated and the exponentiated transmuted models from the statisticalliterature. Further, we present some new models.


Cordeiro [10], pointed out a interesting physical interpretation of the EG class of distributions whichcan be extended to the ET class of distributions whenever α and β are integers. If we consider a paral-lel system with β independent components, each components being a series system with α componentsindependent identically distributed according to a transmuted baseline distribution G(x) and same trans-mutation parameter λ, denoted by U(x), the lifetime of the whole system is an ET distribution.

So, ET class of distributions has also a similar interpretation, the new element introduced, the trans-mutation of the underlying cdf G(x) can be justified by the better fittings of the transmuted distributionover the baseline distribution. Recently, in the statistical literature many distributions have been general-ized using the transmutation map: the exponentiated transmuted generalized Rayleigh distribution (2015)[3], the exponentiated transmuted Weibull distribution (2014) [1], the exponentiated Gumbel (2005)[23],exponentiated power Lindley distribution (2014) [29], exponentiated generalized inverted exponentialdistribution (2014) [25], the transmuted Gamma distribution (2015)[28], the transmuted Weibull distri-bution (2011) [5], the transmuted log-logistic distribution (2013)[6], the transmuted modified Weibulldistribution (2013)[16], the transmuted Weibull Lomax distribution (2015) [2].

This paper is organized as follows. In Section 2, we introduce the exponentiated generalized trans-muted class of distributions with its useful expansion of the density function and its mathematical prop-erties: moments, generating function, Lorenz curve, Bonferroni index, mean and median deviations. Wealso give expansions of the quantile function based on the quantile function of the parent distribution.We derive the entropies: Shannon and Renyi and the order statistics. In Subsection 2.6 we proof someinteresting characteristics of the order statistics and namely we proof that the asymptotic distributionsassociated with the extrem order statistics do not depend on the transmutation parameter λ, but only onthe shape parameters α and β. In Subsection 2.7 we discuss the maximum likelihood estimation and inSubsection 2.8 the stochastic interpretation. In Sections 3 we present some new submodels. In Section 4we performe data analysis and in the last section we conclud this paper.

2 The Exponentiated Transmuted Class of distributions (ET)

The ET class of distributions by the following cumulative distribution function and corresponding densityfunction.

Definition 1. Given an arbitrary continuous distribution defined by its cdf G(x) and pdf g(x), we define

FET(x) = [1 − (1 − {(1 + λ)G(x) − λG2(x)})α]β (1)

where α > 0, β > 0 are two addtional shape, |λ| ≤ is the transmutation parameter.and the corresponding density function has the form

fET(x) = αβ{1−{(1+λ)G(x)−λG2(x)}}α−1[1−{1−{(1+λ)G(x)−λG2(x)}}α]β−1g(x)[1+λ−2λG(x)

](2)

This new class of distributions has many submodels, some known, but many unknown.

(i) For α = 1, we obtain the exponentiated transmuted class of distributions [1, 3]

(ii) For λ = 0, we get the exponentiated generalized class of distribution [10]


(iii) For α = β = 1, we get the transmuted class of distributions [2, 28]

(iv) For α = β = 1 and λ = 0, we obtain the baseline distribution

(v) For α = 1 and λ = 0, we get the exponentiated class of distributions [13, 24]

(vi) For β = 1 and λ = 0, we obtain the Lehmann type II distribution [19]

2.1 Useful Expansions

In this section we present two useful representations of the density function fET(x) given by 2.The first expansion of the density function is

Theorem 1.

fET(x) = αβg(x)[1 + λ − 2λG(x)

] ∞∑j=0

t j

[(1 + λ)G(x) − λG2(x)

] j(3)

where the coefficients t j = t j(α, β) are

t j =(−1) jΓ(β)

j!

∞∑k=0

(−1)kΓ((k + 1)α)Γ(β − k)Γ((k + 1)α − j)k!

Proof. For any real non-integer β, we consider the power series expansion [26]

(1 − z)β−1 =

∞∑k=0

(−1)kΓ(β)Γ(β − k)k!

zk, |z| < 1 (4)

Using the binomial of Newton we have the following

[(1 + λ)G(x) − λG2(x)

] j=

j∑k=0

Ckj(−1)k(1 + λ)kλ j−kG2 j−k(x) (5)

�

Further, (3) can be rewritten as

Theorem 2.

fET(x) =

∞∑j=0

t∗jh j+1(x) (6)

where h j+1(x) = ( j+1)g(x)G j(x) is the exponentiated-G distribution Exp j+1(G), also called the Lehmanntype I distribution [10] and


t∗j = t∗j(α, β, λ) =

t(1)0,0 , j = 0

t(1)1,1 − t(2)

0,0 , j = 1

t(1)2,2 + t(1)

1,0 − t(2)1,1 , j = 2

t(1)3,3 + t(1)

2,1 − t(2)2,2 − t(2)

1,0, , j = 3

t(1)j, j +

j/2−1∑k=0

(t(2)

j−1−k, j−2−2k − t(2)j−1−k, j−1−2k

), j = 2p, p ∈N, j > 2

t(1)j, j +

( j+1)/2−2∑k=0

(t(1)

j−1−k, j−2−2k − t(2)j−1−k, j−1−2k

)− t(2)

( j+1)/2−1,0 , j = 2p + 1, p ∈N, j > 3

t(1)j,k = αβt j

j∑k=0

(−1)kCkj(1 + λ)k+1λ j−k

2 j − k + 1

t(2)j,k = αβt j

j∑k=0

2(−1)kCkj(1 + λ)kλ j−k+1

2 j − k + 2

2.2 Moments, Generating function, Lorenz curve and Bonferroni index

2.2.1 Moments

Let X be a random variable with its cumulative distribution function having the form (1) and W j bea random variable with h j+1(x) = ( j + 1)g(x)G j(x) as its density distribution function and H j+1(x) =

[G(x)] j+1 the corresponding cumulative distribution function. Using the expansion (3) of the densityfunction of X we derive the following formulae for the moments.

Theorem 3.

EFET E(Xi) =

∞∑j=0

t∗j(α, β, λ)E(Wij) (7)

where E(Wij) are the ith moments of the random variable W j.

The moments of W j can be rewritten in terms of the quantile function of G.

Proposition 1.

E(Wij) =

∫ 1

0( j + 1)qG(u)iu jdu (8)

where qG(x) = G−1(x) is the quantile function of G.


2.2.2 Mean and median deviations

The mean and median deviations (σ1(X) and σ2(X), respectively) can be obtained in terms of the firstincomplete moment. So,

σ1(X) = 2µFET(µ) − 2m1(µ) and σ2(X) = µ − 2m1(M) (9)

respectively, where M = qG

([1−(1−2−β)1/α]

)is the median of X, µ = E[X] and m1(z) =

∫ z−∞

x f (x)dxis the first incomplete moment.

Using the expansion of the density function of the ET class of distributions, we derive another for-mula for σ1(X) and σ2(X).

Theorem 4. The mean and median deviations are

σ1(X) = 2µF(µ) − 2∞∑j=0

t∗jU j+1(µ) and σ2(X) = µ − 2∞∑j=0

t∗jU j+1(M)

where

U j+1(x) =

∫ x

−∞

yh j+1(y)dy (10)

is the first incomplete moment of W j.

2.2.3 Generating Function

The first formula for the generating function MX(s) = E[esX] of the ET class of distributions is

Theorem 5.

MX(s) =

∞∑k=0

µk

k!sk (11)

where µk = E[Xk] can be obtained using (7) and (8).

The next formula uses the expansion of the density function for the new family of distributions

Theorem 6.

MX(s) =

∞∑j=0

t∗j(α, β, λ)ϕ j+1(s) (12)

where ϕ j+1(s) is the generating function of the Exp j+1(G) distribution.

The final formula can also be derived from the expansion of density function as

Theorem 7.

MX(s) =

∞∑j=0

t∗j( j + 1)ρ j(s) (13)

where


ρ j(s) =

∫∞

−∞

esxG(x) jg(x)dx =1

j + 1MW j(s)

=

∫ 1

0exp[sqG(u)]u jdu

with qG(u) = G−1(u) quantile function of the parent distribution and MW j(s) the moment generatingfunction of W j.

The Lorenz curve for a random variable X with a continuous cdf F(x) is defined as L(F(x)) =m1(x)E(X)

.

So, using the above formulae we obtain

Theorem 8.

L(FET(x)) =

∞∑j=0

t∗jU j+1(x)

E(Y)

Y being a random variable with a cdf of the form (1).

The formula for the Bonferroni index is obtain in a similar way

Theorem 9.B =

∫∞

0r(y) fET(y)dy

where r(y) =E(X) −m(y)

E(X)and m(y) =

1FET(y)

∫ y

0udFET(u) =

1FET(y)

∞∑j=0

t∗jT j+1(y), T j+1(x) =∫ x0 uh j+1(u)du.

2.3 Quantile Function

The quantile function of the ET class of distribution can be obtained in terms of the quantile function ofthe baseline distribution G(x), by inverting (1). We have for λ , 0

x = QFET (u) = QG

[1 + λ −

√(1 + λ)2 − 4λu2λ

](14)

where u = [1 − (1 − t)α]β, t ∈ (0, 1) and QG(u) = G−1(u) is the quantile function of G.For λ = 0 we have

x = QFET (u) = QG

[1 − (1 − u1/β)1/α

], u ∈ (0, 1) (15)

When the function QG(u) does not have a closed-form expression, it can be expressed as a powerseries


QG(u) =

∞∑i=0

eiui

where the coefficients ei are suitably chosen real numbers. Several well-know distributions in the statis-tical literature, such as the normal, Student-t or gamma distributions, do not have a closed-form of thequantile function. For the standard normal distribution, the coefficients ei are given by

ei = (2π)i/2∞∑

m=i

(−

12

)m−i(mi

)ai

where the quantities ai are defined by ai = 0 for i = 0, 2, 4, ... and ai = p(i−1)/2 for i = 1, 3, 5, ..., andpk satisfy the recursive relations

pk+1 =1

2(2k + 3)

k∑l=0

(2l + 1)(2k − 2l + 1)plpk−l

(l + 1)(2l + 1), p0 = 1, p1 = 1/6, p2 = 7/120, p3 = 127/7560, ...

For λ , 0, λ , −1 we have the following expansion of the quantile function

QFET (u) =

∞∑i=0

i∑k=0

∞∑v,n,m=0

(−1)k+v+m+nΓ(k/2 + 1)Cik

Γ(k/2 + 1 − v)v!

(vβn

)(αnm

)2v(2λ)v−i(1 + λ)i−2veium (16)

For λ = −1 we have

QFET (u) =

∞∑i,m,n=0

((βi)/2

n

)(αnm

)(−1)m+neium (17)

And for λ = 0 we have

QFET (u) =

∞∑i=0

i∑k=0

∞∑n=0

(k/αn

)Ci

k(−1)n+keiun/β (18)

Here(rk)

=r(r−1)...(r−k+1)

k! is the simbol of Pochhammer [ref].

2.4 Entropies

Let X be a random variable with its cumulative distribution function of the form (1).


Theorem 10. The Shannon entropy of X is

H( fET(x)) = −ln(αβ) + (α − 1)∞∑j=0

∞∑n=1

n∑k=0

Ckn(−1)k(1 + λ)n−kλk

t∗j( j + 1)

n( j + n + k + 1)

+ (β − 1)∞∑

j,k=0

∞∑n=1

k∑l=0

t∗j( j + 1)

n( j + k + l + 1)(−1)k+lΓ(αn + 1)Γ(αn + 1 − k)k!

Clk(1 + λ)k−lλl

−

∞∑j=0

t∗j( j + 1)∂∂a

[ ∫∞

−∞

ga+1(x)G j(x)dx]a=0

− ln(1 + λ) +

∞∑j=0

∞∑n=1

t∗j( j + 1)

n( j + n + 1)

( 2λ1 + λ

)n

(19)

Theorem 11. The Renyi entropy of X is

Jγ(X) =1

1 − γln

{ ∞∑k=0

l∑p=0

∞∑n=0

(−1)k+l+p+nΓ(γ(β − 1) + 1)Γ(αk + γ(α − 1) + 1)Γ(γ(β − 1) + 1 − k)Γ(αk + γ(α − 1) − l)k!l!

× Cpl (1 + λ)l+p+γλp

( 2λ1 + λ

)n Γ(γ + 1)Γ(γ + 1 − n)n!

∫∞

−∞

Gl+p+n(x)gγ(x)dx}

(20)

2.5 Order Statistics

The density fi:n(x) of the ith order statistics, for i = 1,n from independent identically distributed randomvariables Y1, ...,Yn is given by

fi:n =f

B(i,n − i + 1)Fi−1F

n−i

where F(x) = 1 − F(x) is the survival function.

We have the following

Theorem 12.

fi:n(x) =αβ

B(i,n − i + 1)u(x)

∞∑l=0

slU(x)l

=αβ

B(i,n − i + 1)g(x)[(1 + λ) − 2λG(x)]

∞∑l=0

sl

l∑k=0

Ckl (−1)k(1 + λ)kλl−kG(x)2l−k (21)

where

sl = sl(α, β, i,n)

=

n−i∑k=0

∞∑r=0

(−1)k+r+lΓ(β(i + k))Γ(α(r + 1))Γ(n − i + 1)Γ(α(r + 1) − l)Γ(β(i + k) − r)Γ(n − i − k + 1)k!r!l!

and u(x) = g(x)[1 + λ − 2λG(x)

]and U(x) = (1 + λ)G(x) − λG2(x)


(21) can be rewriten as

Theorem 13.

fi:n(x) =

∞∑l=0

s∗l hl+1(x) (22)

where hl+1(x) = (l + 1)g(x)G(x)l is the Exp(G) distribution and

s∗j = s∗j(α, β, λ) =

s(1)0,0 , j = 0

s(1)1,1 − s(2)

0,0 , j = 1

s(1)2,2 + s(1)

1,0 − s(2)1,1 , j = 2

s(1)3,3 + s(1)

2,1 − s(2)2,2 − s(2)

1,0, , j = 3

s(1)j, j +

j/2−1∑k=0

(s(2)

j−1−k, j−2−2k − s(2)j−1−k, j−1−2k

), j = 2p, p ∈N, j > 2

s(1)j, j +

( j+1)/2−2∑k=0

(s(1)

j−1−k, j−2−2k − s(2)j−1−k, j−1−2k

)− s(2)

( j+1)/2−1,0 , j = 2p + 1, p ∈N, j > 3

s(1)l,k =

αβ

B(i,n − i + 1)sl

∑k=0

(−1)kCkl (1 + λ)k+1λl−k

2l − k + 1

s(2)l,k =

αβ

B(i,n − i + 1)sl

l∑k=0

2(−1)kCkl (1 + λ)kλl−k+1

2l − k + 2

2.6 Limiting distributions of order statistics

In this subsection we give some new generalized theorems for order statistics. Order statistics are animportant tool in industry and enginerring, being used for quality control methods, reliability studies andnot only [4]. For example, a k-out-of-n system (a system that works as long as least k out of n componentsare working) has the (n-k+1)th order statistics, Xn−k+1:n, as its survival function. The parallel system is a1-out-of-n system and a series syste is a n-out-of-n system. But, in most of the cases we can not work withthe explicit form of the order statistics. So, given an arbitrary cumulative distribution function G(x) wecan obtain the limiting distributions of order statistics for the new generalized class of distribution (ET)using the following three theorems. This is a useful tool because we only need the baseline distributionto determin the asympotic distributions.

2.6.1 Characterizations of the ET class of distributions

Let

C1 = {F a cdf| limt→0

F(tx)F(t)

< ∞},

C2 = {F a cdf| limt→∞

F(t + x)

F(t)< ∞}


. and

C3 = {F a cdf| limx→∞

ddx

( 1h(x)

)= 0}

where h(x) =f (x)

1−F(x) is the hazard rate function and F(x) = 1 − F(x).

Theorem 14. If G ∈ C1 then it follows that also FET ∈ C1, FET being defined by (1).

Proof. We have that

limt→0

FET(tx)FET(t)

= limt→0

[1 − {1 −U(tx)}α]β

[1 − {1 −U(t)}α]β

where U(t) = (1 + λ)G(t) − λG2(t) and u(t) = g(t)[1 + λ − 2λG(t)].Using L’Hospital we obtain

limt→0

1 − {1 −U(tx)}α

1 − {1 −U(t)}α= lim

t→0

{1 −U(tx)}α−1u(tx)x{1 −U(t)}α−1u(t)

= limt→0

xu(tx)u(t)

We have that

limt→0

U(tx)U(t)

= limt→0

xg(tx)[1 + λ − 2λG(tx)]g(t)[1 + λ − 2λG(t)]

= limt→0

xg(tx)g(t)

= l

where l = limt→0

G(tx)G(t)

< ∞.

So,

limt→0

1 − {1 −U(tx)}α

1 − {1 −U(t)}α= l

and

limt→0

FET(tx)FET(t)

= lβ < ∞

�

An important remark is

Remark 1. Theorem 14 shows that the limit of FET(x) from C1 is dependent only of β, so invariant to αand λ.

Theorem 15. If G ∈ C2 then it follows that also FET ∈ C2.


Proof. Using L’Hospital we have

limt→∞

1 − FET(t + x)1 − FET(t)

= limt→∞

1 − [1 − {1 −U(t + x)}α]β

1 − [1 − {1 −U(t)}α]β

= limt→∞

[1 − {1 −U(t + x)}α]β−1[1 −U(t + x)]α−1u(t + x)[1 − {1 −U(t)}α]β−1[1 −U(t)]α−1u(t)

We have

limt→∞

1 − {1 −U(t + x)}α

1 − {1 −U(t)}α= lim

t→∞

u(t + x)[1 −U(t + x)]α−1

u(t)[1 −U(t)]α−1

Because G ∈ C2 we get

limt→∞

1 −U(t + x)1 −U(t)

= limt→∞

g(t + x)[1 + λ − 2λG(t + x)]g(t)[1 + λ − 2λG(t)]

= limt→∞

g(t + x)g(t)

= l

where l = limt→∞

1 − G(t + x)1 − G(t)

< ∞.

Hence, using the following

limt→∞

1 − {1 −U(t + x)}α

1 − {1 −U(t)}α= l lim

t→∞

[1 −U(t + x])α−1

[1 −U(t)]α−1= lα

the proof is complete

limt→∞

1 − FET(t + x)1 − FET(t)

= lαβ < ∞

�

Remark 2. Theorem 15 shows that the limit of FET(x) from C2 is dependent only of α and β, so invariantto λ.

Theorem 16. If G ∈ C3 then it follows that also FET ∈ C3

Proof. We can write FET(x) =[1−(1−K(x))α

]β=

∞∑j=1

w jU j(x) and fET(x) = αβu(x)[1−(1−U(x))α

](1−

U(x))α−1 = αβu(x)∑j≥1

t jU(x) j

where w j =

∞∑k=0

(−1)k+ jΓ(β + 1)Γ(kα + 1)Γ(β − k)k! j!

and U(x) = (1 + λ)G(x) − λG2(x), u(x) = g(x)[1 + λ −

λG(x)].

Using the above forms for the cdf and pdf, we obtain the following


ddx

( 1hET(x)

)=

ddx

1 − FET(x)f (x)

=ddx

U(x)∑j≥1

w j(1 + U(x) + U2(x) + ... + U j−1(x))

αβu(x)∑j≥0

t jU j(x)

=1αβ

ddx

(U(x)u(x)

)∑j≥1

w j(1 + U(x) + U2(x) + · · ·U j−1(x))∑j≥0

t jU j(x)

+1αβ

U(x)u(x)

ddx

∑j≥1

w j(1 + U(x) + U2(x) + · · ·U j−1(x))∑j≥0

t jU j(x)

We have that

ddx

∑j≥1

w j(1 + U(x) + U2(x) + · · ·U j−1(x))

∑j≥0 t jU j(x)

=u(x)

[∑j≥0

t jU j(x)]2

{∑j≥1

w j(1 + 2U(x) + · · · + ( j − 1)U j−2(x))

× (∑j≥0

t jU j(x)) −∑j≥1

w j(1 + U(x) + U2(x) + · · · + U j−1(x))[t1 + 2t2U(x) + 3t3U2(x) + · · · ]}

Now, using the fact that G ∈ C3 and

ddx

U(x)U(x)

=d

dxG(x)g(x)

[1 − λG(x)][1 + λ − λG(x)]

=ddx

(G(x)g(x)

) 1 − λG(x)1 + λ − λG(x)

−λG(x)[2 + λ − 2λG(x)]

[1 + λ − λG(x)]2

we get that limx→∞

ddx

(U(x)u(x)

)= 0; this concluding the proof.

�

2.6.2 Extreme order statistics

The parallel and series systems are represented by the maximum Xn:n and by the minimum X1:n, of arandom sample X1, ...,Xn. The following theorems give us the asymptotic distributions of order statisticsunder certain conditions.

Theorem 17. Let X1:n and Xn:n be the minimum and maximum of a random sample X1,X2, ...,Xn from(2).


(i) If G ∈ C1 and limt→0

G(tx)G(t)

= xθ1 , for each x > 0, where θ1 > 0, then

limn→∞

P{

X1:n − an

bn≤ x

}= 1 − exp(−xθ1β), x > 0

(ii) If G ∈ C2 and limt→∞

1 − F(t + x)1 − F(t)

= exp(−θ2x) for each x > 0, where θ2 > 0, then

limn→∞

P{a∗n(Xn:n − b∗n) ≤ x

}= exp(−exp(−αβθ2))

(iii) If G ∈ C3 then

limn→∞

P{Xn:n − cn

dn≤ x

}= exp(−exp(−x)), x > 0

where an, bn, a∗n, b∗n, cn, dn are norming constants.

Proof. For (i) we apply Theorem 8.3.3 [4] and Theorem 14 and for (ii), Theorem 1.6.2 [17] and Theorem15.

The last part, (iii), follows from Theorem 8.3.3 [4] and Theorem 16. �

The form of the norming constants can be determined following Corollary 1.6.3 [17] and the resultsfrom [4].

Theorem 18. Let X1:n ≤ X2:n ≤ · · ·Xn:n be the order statistics of a random sample X1,X2, · · ·Xn from(2). We have the following

(i) If G ∈ C1 and limt→0

G(tx)G(t)

= xθ1 for each x > 0, where θ1 > 0, then for each i = 1,n

limn∞

P{

Xi:n − an

bn≤ x

}= 1 −

i−1∑k=0

exp{−θβ1}θ

kθ1β1

k!, x > 0

(ii) If G ∈ C3 then for each i = 1,n

limn→∞

P{

Xn−i+1:n − cn

dn≤ x

}=

i−1∑r=0

exp(−exp(−x))exp(−rx)

r!

where an, bn, cn and dn are norming constants.

Proof. The theorem follows from Eqs. (8.4.2) and (8.4.3) of [4], Theorem 14 and Theorem 16. �


2.7 Maximum Likelihood Estimation

Let x1, ..., xn be a random sample of size n from the ET(α, β, λ, η) distribution, where η is a m vectorof unknown parameters in the parent distribution G(x; η). The log-likelihood function for the vector ofparameters γ = (α, β, λ, ηT)T is

l(γ) = nln(α) + nln(β) +

n∑i=1

ln[g(xi; η)[(1 + λ) − 2λG(xi; η)]]

+ (α − 1)n∑

i=1

ln[1 − [(1 + λ)G(xi; η) − λG2(xi; η)]]

+ (β − 1)n∑

i=1

ln{1 − [1 − [(1 + λ)G(xi; η) − λG2(xi; η)]]α} (23)

The log-likelihood can be maximized by solving the nonlinear likelihood equations obtained bydifferentiating (23). The components of the score vector S(γ) are

Sα(γ) =nα

+

n∑i=1

ln[1 −U(xi; η, λ)

]{1 −

(β − 1)[1 −U(xi; η, λ)]α

1 − [1 −U(xi; η, λ)]α}

Sβ(γ) =nβ

+

n∑i=1

ln{1 − [1 −U(xi; η, λ)]α

}

Sη j(γ) =

n∑i=1

{ [u(xi; η, λ)]η j

u(xi; η, λ)−

(α − 1)[U(xi; η, λ)]η j

1 −U(xi; η, λ)+α(β − 1)[1 −U(xi; η, λ)]α−1[U(xi; η, λ)]η j

1 − [1 −U(xi; η, λ)]α}

Sλ(γ) =

n∑i=1

{ g(xi; η) − 2G(xi; η)g(xi; η)[(1 + λ) − 2λG(xi; η)]

−(α − 1)[G(xi; η) − G2(xi; η)]

1 −U(xi; η, λ)

+α(β − 1)[1 −U(xi; η, λ)]α−1[G(xi; η) − G2(xi; η)]

1 − [1 −U(xi; η, λ)]α}

where [u(xi; η)]η j = [g(xi; η)]η j[(1 + λ)− 2λG(xi; η)]− 2λg(xi; η)[G(xi; η)]η j , [g(xi; η)]η j =∂u(xi;η,λ)∂η j

and [G(xi; η)]η j =∂U(xi;η,λ)

∂η j.

In order to construct confidence intervals and hypothesis tests, we need the observed informationmatrix, J(γ), give in the Appendix. The asymptotic distribution of

√n(γ − γ) is Nm+3(0, I(γ)−1), where

I(γ) is the expected information matrix and m is the number of parameters of the baseline distributionG(x), under some mild conditions. In practice, the expected information matrix can be replaced bythe observed information matrix J(γ) evaluated at γ. Using the normal asymptotic distribution one canconstruct approximate confidence intervals and regions.


2.8 Stochastic interpretation

Stochastic ordering is used in portofolio literature, decision theory and also in reliability. Bapat et al.[7] have showed under mild conditions that if X1, ...,Xn are independent and likelihood-ratio orderedthen their order statistics are also likelihood-ratio ordered. In this section we show that the ET family ofdistributions can be ordered in terms of the likelihood ratio order.

Definition 2. [27] Let X1 and X2 be continuous random variables with densities f1 and f2, respectively,such that

f2(x)f1(x)

is non-decreasing over the union of the supports of X1 and X2.

Then X1 is said to be smaller than X2 in the likelihood ratio order (denoted by X1 ≤LR

X2).

Remark 3. It is well-known that the likelihood ratio is stronger than the hazard rate oder and thestochastic order, X1 ≤

LRX2 ⇒ X1 ≤

HRX2 ⇒ X1 ≤

STX2 [30], which are defined as follows.

Definition 3. Let X1 and X2 be two random variables with respective cumulative distribution functionsF1 and F2 and the hazard rates h1 and h2, respectively. Then

(i) X1 is said to be stochastically smaller than X2, denoted by X1 ≤ST

X2, if F1(x) ≥ F2(x) for all x.

(ii) X1 is said to be smaller than X2 in the hazard rate order, denoted by X1 ≤HR

X2, if h1(x) ≤ h2(x)

for all x.

The ET class of distributions can be ordered in the following way.

Theorem 19. Let X1 and X2 be two random variables with the same baseline distribution G(x) and den-sities of the form (2), f1(x) and f2(x), respectively, of parameters α1, β1 > 0, |λ1| ≤ 1 and α2, β2, |λ2| ≤ 1,respectively. If α1 ≥ 1, α2 ≤ 1 β1 ≤ 1 ≤ β2 and −1 ≤ λ2 ≤ 0 ≤ λ1 ≤ 1, then X1 ≤

LRX2.

Proof. We have that

f2(x)f1(x)

=α2β2[1 − (1 + λ2)G(x) + λ2G2(x)]α2−1

α1β1[1 − (1 + λ1)G(x) + λ1G2(x)]α1−1

1 + λ2 − 2λ2G(x)1 + λ1 − 2λ1G(x)

×{1 − [1 − (1 + λ2)G(x) + λ2G2(x)]α2}

β2−1

{1 − [1 − (1 + λ1)G(x) + λ1G2(x)]α1}β1−1

and

lnf2(x)f1(x)

= ln(α2β2

α1β1

)+ (α2 − 1)ln{1 − (1 + λ2)G(x) + λ2G2(x)} − (α2 − 1)ln{1 − (1 + λ1)G(x) + λ1G2(x)}

+ (β2 − 1)ln{1 − [1 − (1 + λ2)G(x) + λ2G2(x)]α2} − (β1 − 1)ln{1 − [1 − (1 + λ1)G(x) + λ1G2(x)]α1}

+ ln{1 + λ2 − 2λ2G(x)} − ln{1 + λ1 − 2λ1G(x)}


f2(x)f1(x) is non-decreasing if and only if

(ln f2(x)

f1(x)

)′≥ 0 for all x, where

(ln

f2(x)f1(x)

)′= −(α2 − 1)

(1 + λ2)g(x) − 2λ2G(x)g(x)1 − (1 + λ2)G(x) + λ2G2(x)

+ (α1 − 1)(1 + λ1)g(x) − 2λ1G(x)g(x)1 − (1 + λ1)G(x) + λ1G2(x)

+ α2(β2 − 1)[1 − (1 + λ2)G(x) + λ2G2(x)]α2−1g(x)[1 + λ2 − 2λ2G(x)]

1 − [1 − (1 + λ2)G(x) + λ2G2(x)]α2

− α1(β1 − 1)[1 − (1 + λ1)G(x) + λ1G2(x)]α1−1g(x)[1 + λ1 − 2λ1G(x)]

1 − [1 − (1 + λ1)G(x) + λ1G2(x)]α1−

2λ2g(x)1 + λ2 − 2λ2G(x)

+2λ1g(x)

1 + λ1 − 2λ1G(x)

It is easy to see that α1 ≥ 1, α2 ≤ 1 β1 ≤ 1 ≤ β2 and −1 ≤ λ2 ≤ 0 ≤ λ1 ≤ 1 imply(ln f2(x)

f1(x)

)′≥ 0 for

all x, and the result holds. �

Corollary 1. Let X1 and X2 be two random variables with the same baseline distribution G(x) and den-sities of the form (2), f1(x) and f2(x), respectively, of parameters α1, β1 > 0, |λ1| ≤ 1 and α2, β2, |λ2| ≤ 1,respectively. If α1 ≥ 1, α2 ≤ 1 β1 ≤ 1 ≤ β2 and −1 ≤ λ2 ≤ 0 ≤ λ1 ≤ 1, then

(i) E[Xr1] ≤ E[Xr

2] for all r > 0, r integer.

(ii) h1(x) ≤ h2(x) for all x, where h1(x) and h2(x) are the corresponding hazard rate functions of f1(x)and f2(x), respectively.

Proof. Using the above Remark 3, Theorem 19 and by taking into account that X1 ≤ST

X2 holds if and

only if

E[Φ(X1)] ≤ E[Φ(X2)] for all non-decreasing function Φ,

(i) follows. For (ii), we use Remark 3 and Theorem 19. �

The ET class of distributions can also be used as mixing distribution with various applications inmany fields. In this context, we derived the following theorems concerning the compounding of the ETdistribution with the binomial and negative binomial distributions.

Let H(x; n, α, β, λ) denote the cumulative distribution function of the binomial-ET random variablewith parameters n, α > 0, β > 0, |λ| ≤ 1 and baseline distribution G(x).

Theorem 20. Let Y1 and Y2 be two binomial-ET random variables with cumulative distribution func-tions H(x; n, α1, β1, λ1) and H(x; n, α2, β2, λ2), respectively and same baseline distribution G(x). Ifα1 ≥ 1, α2 ≤ 1 β1 ≤ 1 ≤ β2 and −1 ≤ λ2 ≤ 0 ≤ λ1 ≤ 1, then Y1 ≤

LRY2.

Proof. Let X1 and X2 be two random variables with the same baseline distribution G(x) and densitiesof the form (2), f1(x) and f2(x), respectively, of parameters α1, β1 > 0, |λ1| ≤ 1 and α2, β2, |λ2| ≤ 1,respectively. The cumulative distribution functions of the binomial ET random variables of parametersn, αi, βi and λi, i = 1, 2, are given by


H(x; n, αi, βi, λi) =

∫∞

−∞

K(x; n, p) fi(p)dp, i = 1, 2

where K(x; n, p) is the cdf of a binomial random variable with parameters n and p. We know thatα1 ≥ 1, α2 ≤ 1 β1 ≤ 1 ≤ β2 and −1 ≤ λ2 ≤ 0 ≤ λ1 ≤ 1, then from Theorem 19 X1 ≤

LRX2. On the

other hand, it is well-known [11] that

Bi(n, p1) ≤LR

Bi(n, p2) whenever p1 ≤ p2

So, the binomial distribution of parameter p is totally positive of order two (TP2). Following Theorem1.C.17 in [30] or Proposition 3.3.54 in [11], the proof is complete. �

Corollary 2. Let Y1 and Y2 be two binomial-ET random variable with the cumulative distribution func-tions H(x; n, α1, β1, λ1) and H(x; n, α2, β2, λ2), respectively and with the corresponding hazard rates r1and r2, respectively. If α1 ≥ 1, α2 ≤ 1 β1 ≤ 1 ≤ β2 and −1 ≤ λ2 ≤ 0 ≤ λ1 ≤ 1, then

(i) E[Yr1] ≤ E[Yr

2] for all r > 0, r integer,

(ii) r1(x) ≤ r2(x) for all x

Let L(x; r, α, β, λ) denote the cumulative distribution function of the negative binomial-ET randomvariable with parameters r, α > 0, β > 0, |λ| ≤ 1 and baseline distribution G(x).

Theorem 21. Let Z1 and Z2 be two negative binomial-ET random variables with cumulative distributionfunctions L(x; r, α1, β1, λ1) and L(x; r, α2, β2, λ2), respectively and same baseline distribution G(x). Ifα1 ≥ 1, α2 ≤ 1 β1 ≤ 1 ≤ β2 and −1 ≤ λ2 ≤ 0 ≤ λ1 ≤ 1, then Z1 ≤

LRZ2.

Proof. We know that the negative binomial distribution is TP2 [30]. So using Theorem 19 and Proposi-tion 3.3.54 [11], the conclusion follows. �

Corollary 3. Let Z1 and Z2 be two negative binomial-ET random variable with the cumulative distri-bution functions L(x; r, α1, β1, λ1) and L(x; r, α2, β2, λ2), respectively and with the corresponding hazardrates r1 and r2, respectively. If α1 ≥ 1, α2 ≤ 1 β1 ≤ 1 ≤ β2 and −1 ≤ λ2 ≤ 0 ≤ λ1 ≤ 1, then

(i) E[Zk1] ≤ E[Zk

2] for all k > 0, k integer,

(ii) r1(x) ≤ r2(x) for all x

3 New Submodels

This section is dedicated to the introduction of new submodels of the ET class, namely: the generalizedexponentiated transmuted exponential distribution (ETExpo), the generalized exponentiated transmutedBirnbaum Saunders distribution (ETBS), the generalized exponentiated transmuted Weibull distribution(ETW) and the generalized exponentiated transmuted generalized Gumbel distribution (ETGumbel).


3.1 The Generalized Exponentiated Transmuted Exponential Distribution (ETExpo)

The exponential distribution has the following cdf and pdf, respectively

G(x) = 1 − exp(−γx), g(x) = γexp(−γx), γ > 0, x > 0

Some submodels and new submodels of the ETExpo distributions presented in the statistical literatureare

(i) the exponentiated transmuted exponential distribution [21] (α = 1)

(ii) the exponentiated generalized exponential distribution (λ = 0)

(iii) the transmuted exponential distribution [31] (α = β = 1)

(iv) the exponentiated exponential class of distribution introduced by Gupta in 2001 [14] (α = 1, λ = 0)[14]

The cumulative distribution function of the ETExpo is

F(x) =[1 −

{1 −

((1 + λ)

(1 − exp(−γx)

)− λ

(1 − exp(−γx)

)2)}α]βand its corresponding density function is

f (x) = αβγexp(−γx)(1 + λ − 2λ

(1 − exp(−γx)

)){1 −

((1 + λ)

(1 − exp(−γx)

)− λ

(1 − exp(−γx)

)2)}α−1

×

[1 −

{1 −

((1 + λ)

(1 − exp(−γx)

)− λ

(1 − exp(−γx)

)2)}α]β−1

Figure 3.1: Plots of the ETExpo density function for some parameters. (1) For α = 0.9 and β = 3. (2)For α = 2.3 and λ = 0.7. (3) For λ = −0.5 and β = 2


3.2 The Generalized Exponentiated Transmuted Birnbaum Saunders Distribution (ETBS)

The Birnbaum Saunders distribution, BS(a, b), a, b > 0, has the following cdf and pdf

G(x) = Φ[1a

{(xb

)1/2−

(bx

)1/2}]

g(x) =1

2√

2πab

[(bx

)1/2+

(bx

)3/2]exp

[−

12a2

(xb−

bx− 2

)], x > 0

where Φ(·) is the cdf of the standard normal distribution.

Some submodels and new submodels of the ETBS distributions presented in the statistical literatureare

(i) the exponentiated transmuted Birnbaum Saunders distribution (α = 1)

(ii) the exponentiated generalized Birnbaum Saunders distribution [9] (λ = 0)

(iii) the transmuted Birnbaum Saunders distribution [8] (α = β = 1) [8]

(iv) the exponentiated Birnbaum Saunders distribution [12] (α = 1, λ = 0) [12]

The cumulative distribution function of the ETBS is

F(x) =[1 −

{1 −

((1 + λ)Φ

[1a

{(xb

)1/2−

(bx

)1/2}]− λΦ

[1a

{(xb

)1/2−

(bx

)1/2}]2)}α]βand its corresponding density function is

f (x) = αβ1

2√

2πab

[(bx

)1/2+

(bx

)3/2]exp

[−

12a2

(xb−

bx− 2

)](1 + λ − 2λΦ

[1a

{(xb

)1/2−

(bx

)1/2}])×

{1 −

((1 + λ)Φ

[1a

{(xb

)1/2−

(bx

)1/2}]− λΦ

[1a

{(xb

)1/2−

(bx

)1/2}]2)}α−1

×

[1 −

{1 −

((1 + λ)G(x) − λΦ

[1a

{(xb

)1/2−

(bx

)1/2}]2)}α]β−1

4 Data Analysis

In this section, we perfom data analysis using as the baseline distribution, the Birnbaum Saunders dis-tribution, BS(a, b), a, b > 0. Ahmed Z. Afify et al.[3] have showed that the exponentiated transmutedgeneralized Rayleigh can be a better model fitting than the transmuted generalized Rayleigh, the gener-alized Rayleigh and the Rayleigh distributions. We show using two data sets that the same can be saidabout the exponentiated transmuted Birnbaum Saunders distribution.


Figure 3.2: Plots of the ETBS density function for some parameters. (1) For α = 0.5 and β = 3.3. (2)For α = 3 and λ = 0.7. (3) For λ = −0.3 and β = 0.7

The criteria used is the maximized log-likelihood −2l, the Akaike Information Criterion (AIC) andthe Bayesian Information Criterion (BIC). The model with minimum AIC or BIC is chosen as the bestmodel to fit the data, where

AIC = −2l + 2k, BIC = −2l + kln(n)

with k the number of parameters and n the sample size.The first data set is an uncensored data set corresponding to remission times (in months) of a random

sample of 128 bladder cancer patients [18, 20]. The first data set is as follows: 0.08, 0.20, 0.40, 0.50,0.51, 0.81, 0.90, 1.05, 1.19, 1.26, 1.35, 1.40, 1.46, 1.76, 2.02, 2.02, 2.07, 2.09, 2.23, 2.26, 2.46, 2.54,2.62, 2.64, 2.69, 2.69, 2.75, 2.83, 2.87, 3.02, 3.25, 3.31, 3.36, 3.36, 3.48, 3.52, 3.57, 3.64, 3.70, 3.82,3.88, 4.18, 4.23, 4.26, 4.33, 4.34, 4.40, 4.50, 4.51, 4.87, 4.98, 5.06, 5.09, 5.17, 5.32, 5.32, 5.34, 5.41,5.41, 5.49, 5.62, 5.71, 5.85, 6.25, 6.54, 6.76, 6.93, 6.94, 6.97, 7.09, 7.26, 7.28, 7.32, 7.39, 7.59, 7.62,7.63, 7.66, 7.87, 7.93, 8.26, 8.37, 8.53, 8.65, 8.66, 9.02, 9.22, 9.47, 9.74, 10.06, 10.34, 10.66, 10.75,11.25, 11.64, 11.79, 11.98, 12.02, 12.03, 12.07, 12.63, 13.11, 13.29, 13.80, 14.24, 14.76, 14.77, 14.83,15.96, 16.62, 17.12, 17.14, 17.36, 18.10, 19.13, 20.28, 21.73, 22.69, 23.63, 25.74, 25.82, 26.31, 32.15,34.26, 36.66, 43.01, 46.12, 79.05. While the first data set is from medicine studies, the second dataset [22] used is from industry and it represents the failure time (in weeks) of 50 items put into used attime t = 0. The data are as follows: 0.013, 0.065, 0.111, 0.111, 0.163, 0.309, 0.426, 0.535, 0.684,0.747, 0.997, 1.284, 1.304, 1.647, 1.829, 2.336, 2.838, 3.269, 3.977, 3.981, 4.520, 4.789, 4.849, 5.202,5.291, 5.349, 5.911, 6.018, 6.427, 6.456, 6.572, 7.023, 7.087, 7.291, 7.787, 8.596, 9.388, 10.261, 10.713,11.658, 13.006, 13.388, 13.842, 17.152, 17.283, 19.418, 23.471, 24.777, 32.795, 48.105.

Table 1 and 2 lists the MLEs of the parameters for the ETBS, TBS and BS distributions and thecorresponding standard errors are given in parentheses. The statistics −2l, AIC and Bic evaluated at themaximum likelihood estimates are also given in the tables. From Table 1 and 2, the ETBS distributionhas the lowest AIC and BIC, thus making it a better model fitting than the TBS and BS distributions.

In order to assess if the model is appropriate, the histograms of the data, the empirical cumulativedistribution function of the data, the plots and the cumulative distribution functions of the fitted ETBS,TBS and BS are displayed in Figures 4 and 4.


Table 1: MLE and criteria of comparison for data set 1.

β λ a bETBS 2.8711893 -0.7567543 2.9524821 0.5237489

(1.9800024) (1.07006124) (0.01281136) (0.01262978)TBS - -0.9000855 1.5373294 2.723449

(0.08254419) (0.12661140) (0.35951918)BS - - 1.374248 4.571078

(0.0862118) (0.44620315)log AIC BIC

ETBS -413.1411 834.2822 845.6904TBS -417.3441 840.6882 849.2443BS -430.0418 864.0836 869.7876

Table 2: MLE and criteria of comparison for data set 2.

β λ a bETBS 1.8399131 -0.9477622 4.9974770 0.2091754

(1.39884582) (1.03978811) (0.08902753) (0.03700269)TBS - -1 ( 3.6457983 0.5264336

0.09428863) (0.01626473) (0.01611407)BS - - 2.762107 1.257570

(0.2973095) (0.2720489)log AIC BIC

ETBS -149.4191 306.8381 314.4862TBS -152.5149 311.0299 316.766BS -161.7123 327.4246 331.2487

Figure 4.1: Model fittings for data sets 1 and 2


Figure 4.2: Fittings of the ecdf for data sets 1 and 2

5 Conclusions

The ET class of distributions has various submodels, some of them unknow and with potential to attractthe attention of statistiscians. The study of systems, due to the limiting theorems of the order statisticsobtained in this paper, that gives us the asymptotic distributions of a parallel or series system, is easier.The properties of the ET class of distributions can be easily derive using the expansions of the densityfunction. Using two data sets, one from medicine and one from industry, we showed that the ET class ofdistribution can be a good model fitting.

References

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Appendix

The elements of the observed information matrix J(γ) for the parameters (α, β, λ, η) are

Jα,α = −nα2 + (β − 1)

n∑i=1

[1 −U(xi; η, λ)]α{ln[1 −U(xi; η, λ)]}2

{1 − [1 −U(xi; η, λ)]α}2

Jα,β = −

n∑i=1

[1 −U(xi; η, λ)]αln[1 −U(xi; η, λ)]1 − [1 −U(xi; η, λ)]

Jα,λ(γ) =

n∑i=1

[−

G(xi; η) − G2xi; η1 − [(1 + λ)G(xi; η) − λG2(xi; η)]

{1 −

(β − 1)[1 − [(1 + λ)G(xi; η) − λG(xi; η)]]α

1 − [1 − [(1 + λ)G(xi; η) − λG2(xi; η)]]α}

− (β − 1)ln[1 − [(1 + λ)G(xi; η) − λG2(xi; η)]]

×α(−1)[1 −U(xi; η, λ)]α−1[G(xi; η) − G2(xi; η)][1 − [1 −U(xi; η, λ)]α]

[1 − [1 −U(xi; η, λ)]α]2

−α[1 −U(xi; η, λ)]2α−1[G(xi; η) − G2(xi; η)]

[1 − [1 −U(xi; η, λ)]α]2


Jα,η j = −

n∑i=1

[U(xi; η, λ)]η j

1 −U(xi; η, λ)+ (β − 1)

n∑i=1

[U(xi; η, λ)]η j[1 −U(xi; η, λ)]α−1

{1 − [1 −U(xi; η, λ)]}2

×

{[1 + αln[1 −U(xi; η, λ)]

][1 − [1 −U(xi; η, λ)]

]+ α[1 −U(xi; η, λ)]α

}

Jβ,β = −nβ2

Jβ,η j = αn∑

i=1

[U(xi; η, λ)]η j[1 −U(xi; η, λ)]α−1

{1 − [1 −U(xi; η, λ)]}2

Jβ,λ(γ) =

n∑i=1

α[1 −U(xi; η, λ)]α−1[G(xi; η) − G2(xi; η)]1 − [1 −U(xi; η, λ)]α

Jλ,λ(γ) =

n∑i=1

[−

[g(xi; η) − 2G(xi; η)]2

u(xi; η, λ)−

(α − 1)[G(xi; η) − G2(xi; η)]2

[1 −U(xi; η, λ)]2

−α(α − 1)(β − 1)[G(xi; η) − G2(xi; η)]2[1 −U(xi; η, λ)]α−2[1 − [1 −U(xi; η, λ)]α]

[1 − [1 −U(xi; η, λ)]α]2

−α2(β − 1)[1 −U(xi; η, λ)]2(α−1)[G(xi; η) − G2(xi; η)]2

[1 − [1 −U(xi; η, λ)]α]2

]

Jλ,η j(γ) =

n∑i=1

{ [u(xi; η, λ)]η jλu(xi; η, λ) − [u(xi; η, λ)]η j[u(xi; η, λ)]λ[u(xi; η, λ]2)

−

(α − 1)[[U(xi; η, λ)]η jλ + [U(xi; η, λ)]η j[U(xi; η, λ)]λ

][1 −U(xi; η, λ)]2

+1

[1 − [1 −U(xi; η, λ)]α]2

{α(β − 1)

[(α − 1)[1 −U(xi; η, λ)]α−2(−1)[U(xi; η, λ)]η j[U(xi; η, λ)]λ

+ [1 −U(xi; η, λ)]α−1[U(xi; η, λ)]η jλ

][1 − [1 −U(xi; η, λ)]α

]− α2(β − 1)[1 −U(xi; η, λ)]2(α−1)[U(xi; η, λ)]η j[U(xi; η, λ)]λ

}}


Jη j,ηs =

n∑i=1

[u(xi; η, λ)]η j,ηsu(xi; η) − [u(xi; η, λ)]η j[u(xi; η, λ)]ηs

[u(xi; η, λ)]2

+

n∑i=1

[U(xi; η, λ)]η jηs[1 −U(xi; η, λ)] + [U(xi; η, λ)]η j[U(xi; η, λ)]ηs

[1 −U(xi; η, λ)]2

−

n∑i=1

[1 −U(xi; η, λ)]α−2

1 − [1 −U(xi; η, λ)]

{(α − 1)[U(xi; η, λ)]η j[U(xi; η, λ)]ηs

− [1 −U(xi; η, λ)][U(xi; η, λ)]η jηs

}+ α

n∑i=1

[U(xi; η, λ)]η j[U(xi; η, λ)]ηs[1 −U(xi; η, λ)]2(α−1)

{1 − [1 −U(xi; η, λ)]α}2

where

[U(xi; η, λ)]η jηs = (1 + λ)[G(xi; η)]η jηs − 2λ[G(xi; η)]ηs[G(xi; η)]η j − 2λG(xi; η)[G(xi; η)]η jηs

and

[u(xi; η, λ)]η jηs = [g(xi; η)]η jηs[(1 + λ) − 2λG(xi; η)] − 2λ[g(xi; η)]η j[G(xi; η)]ηs

− 2λ[g(xi; η)]ηs[G(xi; η)]η j − 2λg(xi; η)[G(xi; η)]η jηs


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International Journal of Risk Theorycefair/files/no9.pdf · proaches such as Bootstrap. Further, in 2010 they developed the exact likelihood for the same distribution but under joint