# Estimating the Reliability Function for a Family of  Exponentiated Distributions

Authors: Ajit Chaturvedi; Anupam Pathak
Journal: Journal of Probability and Statistics
 (2014)

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## Abstract

A family of exponentiated distributions is proposed. The problems of estimating the reliability function are considered. Uniformly minimum variance unbiased estimators and maximum likelihood estimators are derived. A comparative study of the two methods of estimation is done. Simulation study is preformed.


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## Body

## 1. Introduction

The reliability functionR(t) is defined as the probability of failure-free operation until time t. Thus, if the random variable (rv) X denotes the lifetime of an item, then R(t)=P(X>t). Another measure of reliability under stress-strength setup is the probability P=P(X>Y), which represents the reliability of an item of random strength X subject to random stress Y.A lot of work has been done in the literature to deal with inferential problems related to various exponentiated distributions. In particular, Mudholkar and Srivastava [1] considered exponentiated Weibull family for analyzing bathtub failure-real data. The exponentiated Weibull family was also applied to the bus-motor-failure data in Davis [2], to head-and-neck cancer clinical trial data in Efron [3] [see [4]], and in analyzing flood data [see [5]]. Jiang and Murthy [6] considered exponentiated Weibull family and illustrated some of its properties by using graphical approach. Nassar and Eissa [7, 8] studied a two-parameter exponentiated Weibull distribution and they gave some of its properties and estimated the parameters by using the maximum likelihood and Bayes methods based on type II censored data. They used the squared-error and linear in exponential (LINEX) loss functions and an informative prior to obtain the Bayes estimates. Pal et al. [9] showed that the failure rate of exponentiated Weibull behaves more like the failure rate of the Weibull distribution than that of the Gamma distribution. They also obtained maximum likelihood estimators, Fisher’s information matrix, and confidence intervals for related parameters. Gupta et al. [10] introduced exponentiated exponential distribution in a series of papers. Gupta and Kundu [11–17], Gupta et al. [18], and Kundu et al. [19] concentrated on the study of the exponentiated exponential distribution. Kundu and Gupta [20] stated that exponentiated exponential distribution is an alternative to the well-known two-parameter Gamma, two-parameter Weibull or two-parameter lognormal distributions. Kundu and Gupta [20] estimated the reliability of the stress-strength model P(X<Y), when X and Y are independent exponentiated exponential random variables. Raqab and Ahsanullah [21] estimated the parameters of the exponentiated exponential based on ordered statistics. Inference for exponentiated Weibull distribution based on record values was made by Raqab [22]. Abdel-Hamid and AL-Hussaini [23] obtained the maximum likelihood estimates of the parameters when step-stress accelerated life testing is applied for exponentiated exponential distribution. Al-Hussaini and Hussein [24, 25] fitted the exponentiated Burr model to the data of breaking strength of single carbon fibers of particular length (data in [26]) in the complete sample case and when type II censoring is imposed on data. SELF and LINEX loss functions are used in the Bayes estimation. Gupta et al. [10] introduced a new distribution, called the exponentiated Pareto distribution. Shawky and Abu-Zinadah [27] obtained maximum likelihood estimators of the different parameters of the exponentiated Pareto distribution. They also considered five other estimation procedures and compared their performances through numerical simulations.The purpose of the present paper is manyfold. We propose a family of exponentiated distributions, which covers as many as ten exponentiated distributions as specific cases. We consider the problems of estimatingR(t) and “P.” Uniformly minimum variance unbiased estimators (UMVUES) and maximum likelihood estimators (MLES) are derived. In order to obtain these estimators, the major role is played by the estimator of the probability density function (pdf) at a specified point, and the functional forms of the parametric functions to be estimated are not needed. It is worth mentioning here that, in order to estimate “P,” in the literature, the authors have considered the case when X and Y follow the same distributions, may be with different parameters. We have generalized the result to the case when X and Y may follow any distribution from the proposed exponentiated family of distributions. Simulation study is carried out to investigate the performance of the estimators.In Section2, we introduce the family of exponentiated distributions. In Section 3, we derive the UMVUES of R(t) and “P.” In Section 4, we obtain the MLES of R(t) and “P.” In Section 5, analysis of a simulated data has been presented for illustrative purposes. Finally, in Section 6, conclusions have been provided.


## 2. The Family of Exponentiated Distributions

Let the rvX follow the distribution having the pdf
(1)f(x;a,α,λ,θ_)=αλg′(x;a,θ_)e-λg(x;a,θ_)(1-e-λg(x;a,θ_))α-1;hhhhhhhhhhhhhhhhhhx>a≥0,α,λ>0.Here,g(x;a,θ_) is a function of X and may also depend on (may be vector-valued) parameter θ_. Moreover, g(x;a,θ_) is a real-valued, monotonically increasing function of X with g(a;a,θ_)=0, g(∞;a,θ_)=∞, and g′(x;a,θ_) denotes the derivative of g(x;a,θ_) with respect to x. Our case is a special model of the beta extended Weibull family proposed by Cordeiro et al. [28] and can be obtained from equation (7) when b=0. We note that (1) represents a family of exponentiated distributions, as it covers the following exponentiated distributions as special cases.(i)
Forg(x;a,θ_)=x and a=0, we get the exponentiated exponential distribution [see [10]].(ii)
Forg(x;a,θ_)=x2 and a=0, it gives exponentiated Rayleigh distribution [see [29]].(iii)
Forg(x;a,θ_)=xδ, δ>0 and a=0, we get the exponentiated Weibull distribution [see [1]].(iv)
Forg(x;a,θ_)=log⁡(1+xδ), δ>0 and a=0, it leads us to pdf of exponentiated Burr distribution [see [24]].(v)
Forg(x;a,θ_)=log⁡(x/a), it turns out to be exponentiated Pareto distribution [see [10]].(vi)
Forg(x;a,θ_)=(1+(x/ξ)), ξ>0 and a=0, it gives us exponentiated Lomax distribution [see [30]].(vii)
Forg(x;a,θ_)=(1+(xξ/ν)), ξ>0, ν>0 and a=0, it turns out to be exponentiated Burr distribution with scale parameter ν [see [31]].(viii)
Forg(x;a,θ_)=xγexp⁡(νx), γ>0, ν>0 and a=0, it gives the exponentiated form of modified Weibull distribution of Lai et al. [32].(ix)
Forg(x;a,θ_)=γexp⁡[(x/γ)ν-1], γ>0, ν>0 and a=0, we get the exponentiated form of a modification of Weibull distribution [see [33]].(x)
Forg(x;a,θ_)=(x-a)+(ν/λ)log⁡((x+ν)/(a+ν)), ν>0, it gives exponentiated form of the generalized Pareto distribution [see [34]].


## 3. UMVUES ofR(t) and “P”

Throughout this section, we assume thatα is unknown, but “a”, λ, and θ_ are known. Let X1,X2,…,Xn be a random sample of size n from (1).Lemma 1.
LetT=∑i=1nlog⁡(1-e-λg(Xi,a,θ_)). Then, T is complete and sufficient for the family of distributions given in (1). Moreover, the pdf of T is
(2)g(t;α,λ,θ_)=αnΓ(n)(-t)n-1eαt;-∞<t<0.Proof.
Denoting byh(x1,x2,…,x2;a,α,λ,θ_) the joint pdf of X1,X2,…,Xn, we have
(3)h(x1,x2,…,x2;a,α,λ,θ_)=(αλ)n∏i=1ng′(xi;a,θ)e-λ∑i=1ng(xi;a,θ)×e(α-1)∑i=1nlog⁡(1-e-λg(xi;a,θ)).
It follows from (3) and Fishers-Neyman factorization theorem [see [35, p. 341]] that T is sufficient for the family of distributions f(x;a,α,λ,θ_). It is easy to see that the rv V=-2αlog⁡(1-e-λg(x;a,θ_)) follows χ2(2) distribution. Thus, from the well-known additive property of Chi-square distribution [see Johnson and Kotz, [36, p. 170]],  -2αT=-2α∑i=1nlog⁡(1-e-λg(xi;a,θ_)) is an χ2(2n) rv and the result follows. Since the distribution of T belongs to exponential family, it is also complete [see [35, p. 341]].The following lemma provides the UMVUES of the powers ofα.Lemma 2.
Forq∈(-∞,∞), the UMVUE of αq is
(4)α^q={Γ(n)Γ(n-q)(-T)-q;q<n,0;otherwise.Proof.
The result follows from Lemma1, Lehmann-Scheffé theorem [see [35, p. 357]], and the fact that
(5)E(-2αT)-q=Γ(n-q)2qΓ(n);q<n.In the following lemma, we provide the UMVUE of the sampled pdf (1) at a specified point “x.”Lemma 3.
The UMVUE off(x;a,α,λ,θ_) at a specified point “x” is
(6)f^(x;a,α,λ,θ_)={-(n-1)λT-1(1-e-λg(x;a,θ_))-1×g′(x;a,θ_)e-λg(x;a,θ_)×[1-T-1log⁡(1-e-λg(x;a,θ_))]n-2;T<log⁡{1-e-λg(x;a,θ_)},0;otherwise.Proof.
SinceT is complete and sufficient for the family of distributions f(x;a,α,λ,θ_), any function H(T) of T satisfying E[H(T)]=f(x;a,α,λ,θ_) will be the UMVUE of f(x;a,α,λ,θ_). To this end, from (1) and Lemma 1,
(7)∫-∞0H(t)αnΓ(n)(-t)n-1eαtdt=αλg′(x;a,θ)e-λg(x;a,θ)(1-e-λg(x;a,θ))α-1,
or
(8)-αn-1Γ(n)∫log⁡(1-e-λg(x;a,θ))∞H[-u+log⁡{1-e-λg(x;a,θ)}]×[-u+log⁡(1-e-λg(x;a,θ))]n-1e-αudu=λ(1-e-λg(x;a,θ))-1g′(x;a,θ)e-λg(x;a,θ).
Equation (8) is satisfied if we choose
(9)H[-u+log⁡(1-e-λg(x;a,θ_))]={(-(n-1)λg′(x;a,θ_)×e-λg(x;a,θ_)(-u)n-2)×([-u+log⁡(1-e-λg(x;a,θ_))]n-1(1-e-λg(x;a,θ_))×[-u+log⁡(1-e-λg(x;a,θ_))]n-1)-1;u>0,0;otherwise,
and the lemma holds.Remark 4.
We can write (1) as
(10)f(x;a,α,λ,θ_)=λ(1-e-λg(x;a,θ_))-1×g′(x;a,θ_)e-λg(x;a,θ_)α×∑i=0∞[log⁡(1-e-λg(x;a,θ_))]iαii!.
Using Lemma 1 of Chaturvedi and Tomer [37] and Lemma 2, the UMVUE of f(x;a,α,λ,θ_) at a specified point “x” is
(11)f^(x;a,α,λ,θ_)=λ(1-e-λg(x;a,θ_))-1×g′(x;a,θ_)e-λg(x;a,θ_)×∑i=0∞[log⁡(1-e-λg(x;a,θ_))]ii!α^i+1=λ(1-e-λg(x;a,θ_))-1g′(x;a,θ_)e-λg(x;a,θ_)×(-T)-1∑i=0n-2Γ(n-1)Γ(n-i-1)(-T)i×[log⁡(1-e-λg(x;a,θ_))]ii!,
or
(12)f^(x;a,α,λ,θ_)={-(n-1)λT-1(1-e-λg(x;a,θ_))-1×g′(x;a,θ_)e-λg(x;a,θ_)×[1-T-1log⁡(1-e-λg(x;a,θ_))]n-2;T<log⁡{1-e-λg(x;a,θ_)},0;otherwise,
which coincides with Lemma 3. Thus, the UMVUE of the power of α can be used to derive the UMVUE of f^(x;a,α,λ,θ_) at a specified point.
In the following theorem, we obtain the UMVUE ofR(t).Theorem 5.
The UMVUE ofR(t) is given by
(13)R^(t)={1-[1-T-1log⁡{1-e-λg(t;a,θ_)}]n-1;mmmmmmmmmmmT<log⁡(1-e-λg(t;a,θ_)),0;mmmmmmmmmmotherwise.Proof.
Let us consider the expected value of the integral∫t∞f^(x;a,α,λ,θ_)dx with respect to T; that is,
(14)∫0∞{∫t∞f^(x;a,α,λ,θ_)dx}h(t;α,λ,θ_)dt=∫t∞[ET{f^(x;a,α,λ,θ_)}]dx=∫t∞{f(x;a,α,λ,θ_)}dx=R(t).
We conclude from (14) that the UMVUE of R(t) can be obtained simply integrating f^(x;a,α,λ,θ_) from t to ∞. Thus, denoting by g-1(·), the inverse function of g(x;a,θ_), from Lemma 3, we obtain(15)R^(t)=-(n-1)T-1∫tg-1{-λ-1log⁡(1-eT)}λ(1-e-λg(x;a,θ_))-1g′(x;a,θ_)e-λg(x;a,θ_)mmmmmmmmmmmmmmmmmm×[1-T-1log⁡(1-e-λg(x;a,θ_))]n-2dx;T<log⁡{1-e-λg(x;a,θ_)}=-(n-1)∫T-1log⁡(1-e-λg(t;a,θ_))1(1-y)n-2dy,and the theorem follows.LetX and Y be two independent rvs following the families of distributions f1(x;a1,α1,λ1,θ_1) and f2(y;a2,α2,λ2,θ_2), respectively. We assume that α1 and α2 are unknown but a1,a2,λ1,λ2,θ_1, and θ_2 are known. Let X1,X2,…,Xn be a random sample of size n from f1(x;a1,α1,λ1,θ_1) and let Y1,Y2,…,Ym be a random sample of size m from f2(y;a2,α2,λ2,θ_2). Let us denote by T=∑i=1nlog⁡(1-e-λ1g(xi;a1,θ_1)) and S=∑j=1mlog⁡(1-e-λ2h(yj;a2,θ_2)). In what follows, we obtain the UMVUE of “P”.Theorem 6.
The UMVUE of “P” is given by(16)P^={1+(m-1)∫10[1-T-1log⁡{1-e-λ1g(h-1(-(1/λ2)log⁡(1-esν)))}]n-1(1-ν)m-2dν;mmmmh-1{-λ2-1log⁡(1-eS)}>g-1{-λ1-1log⁡(1-eT)},1+(m-1)∫log⁡{1-S-1eλ2h(g-1(-(1/λ1)log⁡(1-eT)))}0[1-T-1log⁡{e-λ1g(h-1(-(1/λ2)log⁡(1-esν)))1-e-λ1g(h-1(-(1/λ2)log⁡(1-esν)))}]n-1mmmmmmmmmmmmmmmmmmmm×(1-ν)m-2dν;mmmmh-1{-λ2-1log⁡(1-eS)}<g-1{-λ1-1log⁡(1-eT)}.Proof.
From the arguments similar to those adopted in proving Theorem5, it can be shown that the UMVUE of “P” is given by
(17)P^=∫y=a2∞∫x=y∞f^1(x;a1,α1,λ1,θ_1)f^2(y;a2,α2,λ2,θ_2)dxdy.
Thus, using Lemma 3, (18)P^=-∫y=a2∞[∫x=y∞(n-1)λ1(1-e-λ1g(x;a1,θ_1))-1g′(x;a1,θ_1)e-λ1g(x;a1,θ_1)mmmmmm·(1-T-1log⁡(1-e-λ1g(x;a1,θ_1)))n-2dx]f^2(y;a2,α2,λ2,θ_2)dy=1+(m-1)λ2S-1∫y=max⁡[h-1{-λ2-1log⁡(1-eS)},g-1{-λ1-1log⁡(1-eT)}]∞{1-T-1log⁡(1-e-λ1g(y;a1,θ_1))}n-1mmmmmmmmm·(1-e-λ2h(y;a2,θ_2))-1h′(y;a2,θ_2)e-λ2h(y;a2,θ_2)mmmmmmmmm·[1-S-1log⁡(1-e-λ2h(y;a2,θ_2))]m-2dy.Let us first consider the case when h-1{-λ2-1log⁡(1-eS)}>g-1{-λ1-1log⁡(1-eT)}. In this case, from (18), (19)P^=1+(m-1)∫10[1-T-1log⁡{1-e-λ1g(h-1(-(1/λ2)log⁡(1-eSν)))}]n-1×(1-ν)m-2dν.Now we consider the case when h-1{-λ2-1log⁡(1-eS)}<g-1{-λ1-1log⁡(1-eT)}. In this case, from (18),
(20)P^=1+(m-1)∫log⁡{1-S-1eλ2h(g-1(-(1/λ1)log⁡(1-eT)))}0[1-T-1log⁡{1-e-λ1g(h-1(-(1/λ2)log⁡(1-esν)))}]n-1(1-ν)m-2dν.
The theorem now follows on combining (19) and (20).Corollary 7.
For the case whena1=a2=a, say, λ1=λ2=λ, say, θ_1=θ_2=θ_, say, and g(x;a,θ_)=h(y;a,θ_),
(21)P^={1-∑i=0n-1(-1)i(n-1)!(m-1)!(n-i-1)!(m+i-1)!(ST)i;T>S,1-∑i=0m-2(-1)i(n-1)!(m-1)!(n+i)!(m-i-2)!(TS)i+1,T<S.Remarks 1.
(i)
In the literature, the researchers have dealt with the estimation ofR(t) and “P,” separately. If we look at the proofs of Theorem 5 and Theorem 6, we observe that the UMVUE of the sampled pdf is used to obtain the UMVUES of R(t) and “P.” Thus, we have established interrelationship between the two estimation problems.
(ii)
In the literature, the researchers have derived the UMVUES of “P” for the case when X and Y follow the same distribution (may be with different parameters). We have obtained UMVUES of “P” for all the three situations, when X and Y follow the same distribution having all the parameters same other than α’s, when X and Y have the same distribution with different parameters and when X and Y have different distributions.
(iii)
It follows from Lemma2 that Var⁡(α^)=α2/(n-1)→0asn→∞. Thus, α^ is a consistent estimator of α. Since f^(x;a,α,λ,θ_), R^(t), and P^ are continuous functions of consistent estimators, they are also consistent estimators of f(x;a,α,λ,θ_), R(t), and “P,” respectively.


## 4. MLES ofR(t) and “P” When All the Parameters Are Unknown

Looking at the distributions belonging to the family (1), we notice that, except for exponentiated Pareto and generalized exponentiated Pareto distributions [(V) and (X)], a=0; that is, “a” is known. For these two distributions θ_ contains “a.” Irrespective of the distribution, the MLE of a is a~=X(1)=min⁡1≤i≤nXi. From (3), the log-likelihood function is
(22)L(a,α,λ,θ_∣x_)=nlog⁡α+nlog⁡λ+∑i=1nlog⁡g′(xi;a,θ_)-λ∑i=1ng(xi;a,θ_)+(α-1)×∑i=1nlog⁡(1-e-λg(xi;a,θ_)).
First of all, we replace “a” by X(1) in (22); then we differentiate (22) with respect to all the unknown parameters and equate these differential equations to zero. The MLES of unknown parameters are obtained on solving these equations simultaneously. Let α~, λ~, and θ~_ be the maximum likelihood estimators of α, λ, and θ_, respectively.The following lemma provides The MLE off(x;a,α,λ,θ_) at a specified point “x”.Lemma 8.
The MLE off(x;a,α,λ,θ_) at a specified point “x” is
(23)f~(x;a,α,λ,θ_)=α~λ~g′(x;a~,θ~_)e-λ~g(x;a~,θ~_)(1-e-λ~g(x;a~,θ~_))α~-1.Proof.
The proof follows from (1) and the one-to-one property of the MLE.In the following theorem, we derive the MLE ofR(t).Theorem 9.
The MLE ofR(t) is given by
(24)R~(t)=1-(1-e-λ~g(t;a~,θ~_))α~.Proof.
Using Lemma8 and invariance property of the MLES,
(25)R~(t)=∫t∞f~(x;a,α,λ,θ_)dx=α~λ~∫t∞g′(x;a~,θ~_)e-λ~g(x;a~,θ~_)(1-e-λ~g(x;a~,θ~_))α~-1dx=α~∫(1-e-λ~g(t;a~,θ~_))1(1-u)α~-1du,
and the theorem follows.In the following theorem, we obtain the maximum likelihood estimator of “P.”Theorem 10.
The MLE of “P” is given by
(26)P~=α~1∫01[1-exp⁡{-λ~2h(g-1{-λ~1-1log⁡(1-u)})}]α~2mmmm×uα~1-1du.Proof.
Using Lemma8 and the one-to-one property of the MLES
(27)P~=∫y=y(1)∞∫x=y∞f~1(x;a1,α1,λ1,θ_1)mmmmmmm×f~2(y;a2,α2,λ2,θ_2)dxdy=∫y=y(1)∞[1-R~2(x)]f~1(x;a1,α1,λ1,θ_1)dx=∫y=y(1)∞[1-exp⁡{-λ~2h(x;a~2,θ~2_)}]α~2mmmm×α~1λ~1g′(x;a~1,θ~1_)e-λ~1g′(x;a~1,θ~1_)mmmm×(1-e-λ~1g′(x;a~1,θ~1_))α~-1dx,
and the result follows.Corollary 11.
For the case whena1=a2=a, say, λ1=λ2=λ, say, θ_1=θ_2=θ_, say, and g(x;a,θ_)=h(y;a,θ_),
(28)P~=α~1α~1+α~2.Remarks 2.
(i)
All the comments made under Remarks1 for UMVUES are tenable for MLES.
(ii)
In the present approaches of obtaining UMVUES and MLES, one does not need the expressions ofR(t) and “P.”
(iii)
Since the UMVUES and MLES of powers ofα are obtained under the same conditions, we compare their performances.  For q=-1 the UMVUE and MLE of α are, respectively, α^=(n-1)(-T)-1 and α~=(n)(-T)-1. For these estimators
(29)V(α^)=α2(n-2),V(α~)=n2(n-1)2(n-2).
Hence,
(30)V(α~)-V(α^)=(2n-1)(n-1)(n-2)α2>0.
Thus, the UMVUE of α is more efficient than its MLE. Similarly, we can compare the performances of these estimators for other powers of α.


## 5. Simulation Studies

In order to verify the consistency of estimators, we have drawn samples of sizesn=5, 20 and 30 from (1) with g(x;a,θ_)=x, a=0, α=3, and λ=1. In Figure 1, we have plotted f^(x;α,λ) corresponding to these samples. We conclude that as the sample size increases, the curves of f^(x;α,λ) come closer to the curve of f(x;α,λ), which is plotted in the same figure only for n=30. For n=30, the curves overlap. A similar pattern follows for the curves of f~(x;α,λ).Figure 1
Cures off^(x;α,λ) and f(x;α,λ).For the case whenα is unknown but other parameters are known, we have conducted simulation experiments using bootstrap resampling technique for sample sizes n=5, 10, 20, and 50. The samples are generated from (1), with, g(x;a,θ_)=x, a=0, α=3, and λ=1. For different values of t, we have computed R^(t), R~(t), their corresponding bias, variance, 95% confidence length, and corresponding coverage percentage. All the computations are based on 500 bootstrap replications and results are reported in Table 1.Table 1
Simulation results forR(t).
t
R
(
t
)
n
=
5
n
=
10
n
=
20
n
=
50
R
^
(
t
)
R
~
(
t
)
R
^
(
t
)
R
~
(
t
)
R
^
(
t
)
R
~
(
t
)
R
^
(
t
)
R
~
(
t
)
0.40
0.9642
0.9628
0.9429
0.9726
0.9638
0.9639
0.9598
0.9656
0.964
−0.0014
−0.0213
0.0084
-
4
e
-
04
-
2
e
-
04
−0.0044
0.0014
-
1
e
-
04
0.001071326
0.001390097
0.0003646232
0.0003069262
0.0002067265
0.0002223408
0.0001467748
0.0001442984
0.1211
0.1182
0.0647
0.0668
0.0555
0.0552
0.0477
0.0476
86.6741
92.6193
91.428
93.112
93.5019
93.8349
94.0056
94.1243
0.50
0.9391
0.8576
0.8546
0.8789
0.8769
0.9282
0.9252
0.9399
0.9386
−0.0815
−0.0845
−0.0601
−0.0622
−0.0109
−0.0139
8
e
-
04
-
5
e
-
04
0.009229477
0.008767263
0.007167494
0.006830551
0.001421963
0.001408017
0.0005898798
0.0005745804
0.1912
0.1524
0.2262
0.2086
0.1412
0.1371
0.0958
0.0947
93.9509
94.0809
94.1651
94.4641
94.2804
94.6018
94.5567
94.6561
0.80
0.8330
0.8314
0.8338
0.8355
0.8374
0.8408
0.8417
0.8319
0.8325
−0.0016
7
e
-
04
0.0025
0.0044
0.0077
0.0087
−0.0011
-
5
e
-
04
0.004528085
0.002894221
0.002485577
0.002035995
0.002223323
0.002035827
0.001419582
0.001364689
0.2552
0.2057
0.191
0.1725
0.1796
0.1711
0.1474
0.1446
94.1964
94.3942
94.4812
94.5182
94.6424
94.6596
94.952
94.9534
1.00
0.7474
0.729
0.7499
0.7936
0.7994
0.7279
0.7338
0.7548
0.7568
−0.0184
0.0025
0.0461
0.052
−0.0195
−0.0137
0.0074
0.0093
0.008217478
0.005415503
0.005849085
0.005767851
0.004407818
0.003903546
0.002428885
0.002383391
0.3395
0.2812
0.2359
0.2142
0.2505
0.2407
0.1915
0.1884
94.0502
94.2394
94.5787
94.6276
94.9352
94.9729
95.0586
95.0578
1.50
0.5311
0.4442
0.4933
0.5094
0.532
0.5283
0.5392
0.5256
0.53
−0.087
−0.0378
−0.0218
9
e
-
04
−0.0028
0.0081
−0.0055
−0.0011
0.01260328
0.00618332
0.00520108
0.004460047
0.004379108
0.004317541
0.001922681
0.001875105
0.2803
0.2719
8 0.2721
0.2648
0.26
0.2564
0.1747
0.1738
94.2755
94.5729
94.464
94.653
94.7342
94.7791
95.0374
95.0517
2.00
0.3535
0.4971
0.5442
0.3349
0.3573
0.3429
0.354
0.3485
0.3529
0.1435
0.1906
−0.0187
0.0037
−0.0106
5
e
-
04
−0.005
-
6
e
-
04
0.02667172
0.0416806
0.004913515
0.004757415
0.00239676
0.002330965
0.001298761
0.001284863
0.3125
0.2928
0.2712
0.2765
0.1923
0.1943
0.1442
0.1448
94.3591
94.7634
94.5513
94.7147
94.7592
94.8151
95.0453
95.0583
2.50
0.2266
0.2396
0.2813
0.2699
0.2906
0.2047
0.2131
0.226
0.2295
0.013
0.0547
0.0433
0.064
−0.0219
−0.0135
-
6
e
-
04
0.0029
0.001733384
0.004863281
0.002956535
0.005265369
0.00116556
0.0009073532
0.0004148542
0.0004315035
0.1576
0.1725
0.1312
0.1364
0.1045
0.1075
0.0809
0.0817
94.0777
94.3206
94.2939
94.3724
94.7371
94.7634
95.0835
95.0883
Here, the first row indicates the estimate, the second row indicates the bias, the third row indicates variance, the fourth row indicates 95% bootstrap confidence length, and the fifth row indicates the coverage percentage.In order to estimate “P,” for the case when α1 and α2 are unknown but other parameters are known, we have conducted simulation experiments using bootstrap resampling technique for sample sizes (n,m) = (5, 5), (10, 10), (15, 15), (25, 25), and (30, 30). The samples are generated from (1), with g(x;a1,θ1_)=x,h(y;a2,θ2_)=y, a1=a2=0, λ1=λ2=1, α1=2, and α2=0.5(0.25)1.25. The computations are based on 500 bootstrap replications. We have computed P^, P~, bias, variance, 95% confidence length, and corresponding coverage percentage. The results are presented in Table 2.Table 2
Simulation results for “P.”
(
α
1
,
α
2
)
(1, 0.50)
(1, 0.75)
(1, 1.00)
(1, 1.25)
P
0.6667000
0.5714286
0.500000
0.4444000
(n, m)
P
~
P
^
P
~
P
^
P
~
P
^
P
~
P
^
(5, 5)
0.4160
0.4071
0.5196
0.5198
0.4937
0.4925
0.4503
0.4445
−0.2507
−0.2595
−0.0518
−0.0516
−0.0063
−0.0075
0.0058
1
e
-
04
0.08305567
0.09161194
0.04060813
0.04711423
0.01738566
0.02128741
0.03447488
0.04041304
0.5225
0.5686
0.6963
0.7375
0.4996
0.5469
0.7179
0.7577
92.2467
91.7696
92.6155
91.9685
93.9423
93.6839
93.2283
92.5227
(10, 10)
0.4219
0.4179
0.5324
0.5340
0.5028
0.5029
0.4477
0.4451
−0.2448
−0.2487
−0.0391
−0.0374
0.0028
0.0029
0.0033
6
e
-
04
0.06910139
0.07194417
0.01062704
0.01145625
0.01023848
0.01128198
0.01168588
0.01284482
0.3560
0.3727
0.3524
0.3698
0.3863
0.4050
0.4024
0.4213
93.4513
93.3550
93.5220
93.4724
94.1638
94.1400
93.2875
93.2019
(15, 15)
0.4507
0.4491
0.5601
0.5620
0.5015
0.5016
0.4499
0.4481
−0.2159
−0.2176
−0.0113
−0.0095
0.0015
0.0016
0.0054
0.0037
0.05200412
0.05309595
0.01157199
0.0122409
0.00717329
0.007664479
0.002471958
0.002628193
0.2739
0.2826
0.4016
0.4130
0.3234
0.3340
0.1911
0.1976
93.2657
93.2813
93.8776
93.8404
94.2206
94.1996
94.6752
94.6696
(25, 25)
0.4676
0.4670
0.5625
0.5637
0.4994
0.4994
0.4466
0.4455
−0.1990
−0.1997
−0.0090
−0.0077
-
6
e
-
04
-
6
e
-
04
0.0021
0.001
0.04415965
0.0446212
0.002974049
0.003065954
0.004487199
0.004669894
0.004399884
0.004568917
0.2529
0.2578
0.2048
0.2088
0.2579
0.2630
0.266
0.271
93.8839
93.87700
94.2551
94.2501
94.3152
94.3066
95.5079
95.4975
(30, 30)
0.4869
0.4867
0.5740
0.5753
0.4999
0.4999
0.447
0.4461
−0.1798
−0.1800
0.0026
0.0038
-
1
e
-
04
-
1
e
-
04
0.0025
0.0017
0.03621819
0.03642065
0.003972225
0.004105375
0.003633788
0.003756843
0.00493606
0.005091428
0.2470
0.2511
0.243
0.2468
0.2391
0.243
0.2797
0.2841
95.2267
95.2211
94.5145
94.5023
95.1531
95.1467
95.3624
95.3519
Here, the first row indicates the estimate, the second row indicates the bias, the third row indicates variance, the fourth row indicates 95% bootstrap confidence length, and the fifth row indicates the coverage percentage.In order to demonstrate the application of the theory developed in Section4, we consider the following example. Let the rv X follow exponentiated exponential distribution with pdf
(31)f(x;α,λ,δ)=αλδxδ-1e-λxδ(1-e-λxδ)α-1;x,α,λ,δ>0,
where α, λ, and δ are unknown. Denoting, L1(α,λ,δ∣x), the likelihood, the log-likelihood is given by
(32)log⁡L1(α,λ,δx)=nlog⁡α+nlog⁡λ+nlog⁡δ+(δ-1)∑i=1nlog⁡xi-λ∑i=1nxiδ+(α-1)∑i=1nlog⁡(1-e-λxiδ).
Considering negative log-likelihood, then differentiating it with respect to all unknown parameters, and equating these differential coefficients to zero, from (32),
(33)-nα-∑i=1nlog⁡(1-e-λxiδ)=0,-nλ+∑i=1nxiδ-(α-1)∑i=1nxiδe-λxiδ(1-e-λxiδ)=0,-nδ-∑i=1nlog⁡xi+λ∑i=1nxiδlog⁡xi-(δ-1)λ×∑i=1nxiδe-λxiδlog⁡xi(1-e-λxiδ)=0.The following sample of size 50 is generated from (31), for α=3.5, λ=1, and δ=1 as follows.Sample 1. 1.4904, 0.5808, 2.2326, 0.9756, 2.3062, 1.4692, 1.0433, 2.9527, 0.9241, 2.5901, 1.2021, 1.3022, 3.6532, 0.8146, 1.7083, 0.3126, 2.4925, 0.9462, 0.4557, 2.0120, 1.6554, 1.1651, 0.9703, 1.0305, 0.4618, 0.7465, 1.9249, 1.5763, 1.9127, 0.9742, 2.7401, 1.3686, 1.8546, 2.3296, 1.4526, 1.1719, 4.4671, 1.1218, 1.1901, 0.8248, 3.0091, 0.9511, 1.3308, 2.2145, 2.0216, 1.6740, 1.5620, 2.0339, 0.6882, 3.1335.Assuming that the data represents life spans of items in hours,R(0.35)=0.986, and solving (33) simultaneously, we get α~=3.711821, λ~=1.170381, δ~=1.098677, and R~(0.35)=0.9872.In order to obtain the maximum likelihood estimator of “P,” we have generated one more sample of size 50 from (31), for α=3, λ=1, and δ=1 as follows.Sample 2. 1.2271, 0.9467, 1.9800, 0.6613, 0.4526, 1.5701, 1.0414, 1.9008, 1.2378, 3.7137, 1.4012, 0.5640, 1.6704, 2.0615, 1.4360, 1.6696, 0.7443, 0.9568, 1.6306, 0.2327, 1.0024, 0.8178, 0.7704, 2.1587, 0.8460, 2.3833, 1.0702, 1.1584, 3.1261, 3.7417, 3.0401, 2.4465, 0.6874, 3.2289, 2.4881, 1.6636, 0.4313, 2.4856, 1.2428, 1.5750, 1.1564, 1.4370, 3.1483, 1.1472, 1.4889, 0.8209, 0.6693, 1.2684, 3.2650, 3.2285.For this sample, we haveα~=2.932394, λ~=1.044041, andδ~=1.106363. Using these results (obtained from Samples 1 and 2) and Theorem 10, we get P=0.5384615 and P~=0.5586546.


## 6. Conclusions

We propose a class of distributions, which covers as many as ten exponentiated distributions as special cases. The problems of estimatingR(t) and “P” are considered. UMVUES and MLES are derived. A comparative study of the two methods of estimation is done. The estimators of “P” are derived, which cover the cases when X and Y may follow the same, as well as, different distributions. By estimating the sampled pdf to obtain the estimators of R(t) and “P,” an interrelationship between the two estimation problems is established. Simulation study is performed, and a real-data example is presented.



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*Source: 563093-2014-04-29.xml*