# Statistical Inference of Stress-Strength Reliability of Gompertz Distribution under Type II Censoring

Authors: Z. Karimi Ezmareh; G. Yari
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

This paper develops the problem of estimating stress-strength reliability for Gompertz lifetime distribution. First, the maximum likelihood estimation (MLE) and exact and asymptotic confidence intervals for stress-strength reliability are obtained. Then, Bayes estimators under informative and noninformative prior distributions are obtained by using Lindley approximation, Monte Carlo integration, and MCMC. Bayesian credible intervals are constructed under these prior distributions. Also, simulation studies are used to illustrate these inference methods. Finally, a real dataset is analyzed to show the implementation of the proposed methodologies.


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## Body

## 1. Introduction

The stress-strength reliabilityR=PX>Y is an assessment of the reliability of a component based on its strength X and its stress Y. The idea of a stress-strength reliability was introduced by Birnbaum [1] and spread two years later by Birnbaum and McCarty [2].Recently, the study on reliability has been considered by the authors, which we refer to some recent studies. Qixuan and Wenhao [3] worked on the Bayesian and classical estimation of stress-strength reliability for inverse Weibull lifetime models. Abravesh et al. [4] obtained classical and Bayesian estimation of stress-strength reliability in type II censored Pareto distributions. Akgül et al. [5] presented inferences on stress-strength reliability based on ranked set sampling data in the case of Lindley distribution. Byrnes et al. [6] made a Bayesian inference of R for Burr type XII distribution based on progressively first failure-censored samples. Zhang et al. [7] studied the reliability of generalized Rayleigh distribution under progressive type II censoring.Gompertz distribution which was first proposed by Gompertz [8] is one of the most widely used distributions in the fields of survival, lifetime data, mortality tables, computer, biology, sociology, and marketing [9–12]. Some recent studies on the distribution of Compretz include the following: [13] presented a new and practical generalization of the Compretz distribution. [14] developed acceptance sampling plans for lot sentencing in which the quality characteristic of the products follows the Topp-Leone Gompertz distribution. An application of Gamma-Gompertz distribution was proposed by [15]. [16] estimated the parameters of a new generalization of Gompertz distribution and investigated the features and application of this new model.The study on reliability by considering Gompertz distribution is one of the most important and interesting issues. Saraçoğlu and Kaya [17] studied MLE and confidence intervals of system reliability for Gompertz distribution in stress-strength models. Kumar and Vaish [18] presented a study of strength reliability for Gompertz distributed stress. Jha et al. [19] obtained reliability estimation of a multicomponent stress-strength model for unit Gompertz distribution under progressive type II censoring. Asadi et al. [20] studied inference on adaptive progressive hybrid censored accelerated life test for Gompertz distribution.A brief explanation of the type II censoring is given. Letx1,x2,⋯,xn1 and y1,y2,⋯,yn2 be independent random samples from X and Y random variable, respectively. Suppose the ordered statistics of these samples are x1<x2<⋯<xn1 and y1<y2<⋯<yn2. xi’s and yi’s are collected until r1 failures and r2 failures occur, respectively (where r1≤n1 and r2≤n2).The rest of the article is organized as follows. In Section2, we introduce the Gompertz distribution. In Section 3, we obtain the MLE of stress strength reliability (R). In Section 4, we construct the exact and asymptotic confidence intervals for R. In Section 5, we calculate the Bayes estimator of R by considering the conjugate informative and Jeffreys noninformative prior distributions. In Section 6, we provide Bayesian credible intervals, including equi-tailed and HPD intervals under the conjugate informative and Jeffreys noninformative prior distributions. In Section 7, the performance of these inference methods is compared by using simulation studies. Finally, Section 8 performs a real data analysis to demonstrate the application of these methods.


## 2. Gompertz Distribution

LetX~Gompertzβ1,γ and Y~Gompertzβ2,γ be two independent random variables. The probability density function (PDF) and cumulative distribution function (CDF) of X and Y are given:
(1)fXx;β1,γ=β1eγxe−β1/γeγx−1,FXx;β1,γ=1−e−β1/γeγx−1,x>0,β1,γ>0,fYy;β2,γ=β2eγye−β2/γeγy−1,FYy;β2,γ=1−e−β2/γeγy−1,y>0,β2,γ>0.The reliability function is calculated as follows:(2)R=PX>Y=∫−∞∞FYwfXwdw=∫0∞1−e−β2/γeγw−1β1eγwe−β1/γeγw−1dw=β2β1+β2.


## 3. MLE ofR

Letx1,x2,⋯,xr1 be a type II censored sample from Gompertzβ1,γ and y1,y2,⋯,yr2 be a type II censored sample from Gompertzβ2,γ. Suppose these two samples are independent. The likelihood function is given by
(3)Lβ1,β2,γx,y=n1!n2!n1−r1!n2−r2!∏i=1r1fXxi;β1,γSXxr1n1−r1×∏j=1r2fYyj;β2,γSYyr2n2−r2=n1!n2!n1−r1!n2−r2!β1r1eγ∑i=1r1xie−β1/γ∑i=1r1eγxi−1e−β1n1−r1/γeγxr1−1×β2r2eγ∑j=1r2yje−β2/γ∑j=1r2eγyj−1e−β2n2−r2/γeγyr2−1=n1!n2!n1−r1!n2−r2!β1r1β2r2eγ∑i=1r1xi+∑j=1r2yje−β1b1′e−β2b2′,where
(4)b1′=1γ∑i=1r1eγxi+n1−r1eγxr1−n1,(5)b2′=1γ∑j=1r2eγyj+n2−r2eγyr2−n2.Then, the log-likelihood function is(6)lβ1,β2,γx,y=logn1!n2!−logn1−r1!n2−r2!+γ∑i=1r1xi+∑j=1r2yj+r1logβ1+r2logβ2−β1γ∑i=1r1eγxi+n1−r1eγxr1−n1−β2γ∑j=1r2eγyj+n2−r2eγyr2−n2.To obtain the MLE of parametersβ1,β2 and γ, it is sufficient to derive the log-likelihood function with respect to parameters β1,β2 and γ and equal them to zero:
(7)∂l∂β1=r1β1−1γ∑i=1r1eγxi+n1−r1eγxr1−n1=0,(8)∂l∂β2=r2β2−1γ∑j=1r2eγyj+n2−r2eγyr2−n2=0,(9)∂l∂γ=r1γ+r2γ+∑i=1r1xi+∑j=1r2yj−β1γ∑i=1r1xieγxi+n1−r1xr1eγxr1−β2γ∑j=1r2yjeγyj+n2−r2yr2eγyr2.From Equations (7) and (8), we get
(10)β^1=γr1∑i=1r1eγxi+n1−r1eγxr1−n1=r1b1′,(11)β^2=γr2∑j=1r2eγyj+n2−r2eγyr2−n2=r2b2′.Now, by substituting (10) and (11) into (9), the MLE of parameter γγ^ is obtained. Then, to get β^1 and β^2, we substitute γ^ into Equations (10) and (11). Therefore, the MLE of R is
(12)R^=β^2β^1+β^2.


## 4. Confidence Interval ofR

In this section, the exact and asymptotic confidence intervals forR are calculated.


### 4.1. Exact Confidence Interval

Letx1,x2,⋯,xr1 be a type II censored sample from Gompertzβ1,γ. Consider Wi=β1/γeγxi−1,i=1,2,⋯,r1, where w1≤w2≤⋯≤wr1 is a type II censored dependent sample from the standard exponential distribution (SED). Now, apply the following conversion:
(13)Wi′=n1−i+1Wi−Wi−1.It can be concluded thatW1′,W2′,⋯,Wr1′~indSED. Therefore,
(14)2∑i=2r1Wi′~χ2r1−12.Similarly, supposey1,y2,⋯,yr2 be a type II censored sample from Gompertzβ2,γ. Define Mj=β2/γeγyj−1,j=1,2,⋯,r2, where m1≤m2≤⋯≤mr1 is a type II censored dependent sample from the SED. Now, apply the following transformation:
(15)Mj′=n2−j+1Mj−Mj−1.It results thatM1′,M2′,⋯,Mr2′~indSED. So,
(16)2∑j=2r2Mj′~χ2r2−12.Based on the independence of∑i=2r1Wi′ and ∑j=2r2Mj′ can be written
(17)F0=r2−1∑i=2r1Wi′r1−1∑j=2r2Mj′~F2r1−1,2r2−1.Then, confidence interval forR is
(18)11+Q2/Q1F1−ξ/22r1−1,2r2−1,11+Q2/Q1Fξ/22r1−1,2r2−1,where
(19)Q1=r2−1γ∑i=2r1n1−i+1eγxi−eγxi−1,Q2=r1−1γ∑j=2r2n2−j+1eγyj−eγyj−1.


### 4.2. Asymptotic Confidence Interval

In this section, the asymptotic confidence interval forR is calculated using Wald statistics. Based on Wald statistics, we have
(20)W=1/r1+1/r2−1/2R^−Rη^⟶DN0,1,where η^=VarR^=β^1β^2/β∧1+β∧22.Theorem 1.
Letr1⟶∞ and r2⟶∞, then
(21)21r1+1r2−1/2R^−R⟶DN0,κ2,where κ2=2β1β22/β1+β22·Proof.
Given that the maximum likelihood estimator is asymptotically normal [21], when r1⟶∞ and r2⟶∞, then
(22)r1β∧1−β1,r2β∧2−β2T⟶DN0,ϖ,where
(23)ϖ=β1200β22.Define hβ1,β2=β2/β1+β2. According to Taylor expansion hβ^1,β^2 around β1 and β2, we have
(24)R^=hβ^1,β^2=hβ1,β2+∇hβ1,β2Tβ^1−β1β^2−β2+ς1=R+−β2β1+β22β1β1+β22·β^1−β1β^2−β2+ς1,a.s.,where the remaining sentences in the following relation apply:
(25)ς1=OPβ∧1−β12+β∧2−β22.
Based on (22) and (24), when r1⟶∞ and r2⟶∞, then
(26)E21r1+1r2−1/2R^−R=0,(27)Var21r1+1r2−1/2R^−R=21r1+1r2−1VarR^−R≔21r1+1r2−1S2,where
(28)S2=Var−β2β1+β22β1β1+β22·β^1−β1β^2−β2=−β2β1+β22β1β1+β22·Varβ^1−β1β^2−β2·−β2β1+β22β1β1+β22=−β2β1+β22β1β1+β22·Varr1β^1−β1r100Varr2β^2−β2r2·−β2β1+β22β1β1+β22=−β2β1+β22β1β1+β22·β12r100β22r2·−β2β1+β22β1β1+β22=κ2r1+κ2r2=κ21r1+1r2.
Therefore,(29)Var21r1+1r2−1/2R^−R=κ2.
Equation (26) holds because according to the property of MLE R^⟶PR.Corollary 2.
A1001−ξ% asymptotic confidence interval for R is
(30)R^−η^zξ/21r1+1r2,R^+η^zξ/21r1+1r2,where η^=β^1β^2/β∧1+β∧22·Proof.
Defineη2=β1β22/β1+β24 and η∧2=β∧1β∧22/β∧1+β∧24. According to the asymptotic property of the MLE, η^/η tends to 1 in probability. On the other hand, according to Theorem 1,
(31)1/r1+1/r2−1/2R^−Rη⟶DN0,1.
Therefore, according to the Slutsky theorem, we have(32)1/r1+1/r2−1/2R^−Rη^=1/r1+1/r2−1/2R^−R/ηη^/η⟶DN0,1.
Thus,(33)PR^−η^zξ/21r1+1r2≤R≤R^+η^zξ/21r1+1r2=1−ξ.


## 4.1. Exact Confidence Interval

Letx1,x2,⋯,xr1 be a type II censored sample from Gompertzβ1,γ. Consider Wi=β1/γeγxi−1,i=1,2,⋯,r1, where w1≤w2≤⋯≤wr1 is a type II censored dependent sample from the standard exponential distribution (SED). Now, apply the following conversion:
(13)Wi′=n1−i+1Wi−Wi−1.It can be concluded thatW1′,W2′,⋯,Wr1′~indSED. Therefore,
(14)2∑i=2r1Wi′~χ2r1−12.Similarly, supposey1,y2,⋯,yr2 be a type II censored sample from Gompertzβ2,γ. Define Mj=β2/γeγyj−1,j=1,2,⋯,r2, where m1≤m2≤⋯≤mr1 is a type II censored dependent sample from the SED. Now, apply the following transformation:
(15)Mj′=n2−j+1Mj−Mj−1.It results thatM1′,M2′,⋯,Mr2′~indSED. So,
(16)2∑j=2r2Mj′~χ2r2−12.Based on the independence of∑i=2r1Wi′ and ∑j=2r2Mj′ can be written
(17)F0=r2−1∑i=2r1Wi′r1−1∑j=2r2Mj′~F2r1−1,2r2−1.Then, confidence interval forR is
(18)11+Q2/Q1F1−ξ/22r1−1,2r2−1,11+Q2/Q1Fξ/22r1−1,2r2−1,where
(19)Q1=r2−1γ∑i=2r1n1−i+1eγxi−eγxi−1,Q2=r1−1γ∑j=2r2n2−j+1eγyj−eγyj−1.


## 4.2. Asymptotic Confidence Interval

In this section, the asymptotic confidence interval forR is calculated using Wald statistics. Based on Wald statistics, we have
(20)W=1/r1+1/r2−1/2R^−Rη^⟶DN0,1,where η^=VarR^=β^1β^2/β∧1+β∧22.Theorem 1.
Letr1⟶∞ and r2⟶∞, then
(21)21r1+1r2−1/2R^−R⟶DN0,κ2,where κ2=2β1β22/β1+β22·Proof.
Given that the maximum likelihood estimator is asymptotically normal [21], when r1⟶∞ and r2⟶∞, then
(22)r1β∧1−β1,r2β∧2−β2T⟶DN0,ϖ,where
(23)ϖ=β1200β22.Define hβ1,β2=β2/β1+β2. According to Taylor expansion hβ^1,β^2 around β1 and β2, we have
(24)R^=hβ^1,β^2=hβ1,β2+∇hβ1,β2Tβ^1−β1β^2−β2+ς1=R+−β2β1+β22β1β1+β22·β^1−β1β^2−β2+ς1,a.s.,where the remaining sentences in the following relation apply:
(25)ς1=OPβ∧1−β12+β∧2−β22.
Based on (22) and (24), when r1⟶∞ and r2⟶∞, then
(26)E21r1+1r2−1/2R^−R=0,(27)Var21r1+1r2−1/2R^−R=21r1+1r2−1VarR^−R≔21r1+1r2−1S2,where
(28)S2=Var−β2β1+β22β1β1+β22·β^1−β1β^2−β2=−β2β1+β22β1β1+β22·Varβ^1−β1β^2−β2·−β2β1+β22β1β1+β22=−β2β1+β22β1β1+β22·Varr1β^1−β1r100Varr2β^2−β2r2·−β2β1+β22β1β1+β22=−β2β1+β22β1β1+β22·β12r100β22r2·−β2β1+β22β1β1+β22=κ2r1+κ2r2=κ21r1+1r2.
Therefore,(29)Var21r1+1r2−1/2R^−R=κ2.
Equation (26) holds because according to the property of MLE R^⟶PR.Corollary 2.
A1001−ξ% asymptotic confidence interval for R is
(30)R^−η^zξ/21r1+1r2,R^+η^zξ/21r1+1r2,where η^=β^1β^2/β∧1+β∧22·Proof.
Defineη2=β1β22/β1+β24 and η∧2=β∧1β∧22/β∧1+β∧24. According to the asymptotic property of the MLE, η^/η tends to 1 in probability. On the other hand, according to Theorem 1,
(31)1/r1+1/r2−1/2R^−Rη⟶DN0,1.
Therefore, according to the Slutsky theorem, we have(32)1/r1+1/r2−1/2R^−Rη^=1/r1+1/r2−1/2R^−R/ηη^/η⟶DN0,1.
Thus,(33)PR^−η^zξ/21r1+1r2≤R≤R^+η^zξ/21r1+1r2=1−ξ.


## 5. Bayesian Estimation ofR

Bayes [22] and Laplace [23] found that the uncertainty about the parameters of a model, which we represent with θ, could be modeled on Θ through a probability distribution such as πθ, called the prior distribution. With this approach, the inference is based on the conditional distribution θ on x, πθx. This conditional distribution is called the posterior distribution. In this section, Bayesian estimation is obtained by using the conjugate informative and Jeffreys noninformative prior distributions.


### 5.1. Conjugate Informative Prior

Letβ1~Gammaα1,η1 and β2~Gammaα2,η2 and are independent. The PDF of these priors is as follows:
(34)πβ1=η1α1Γα1β1α1−1e−η1β1,α1,η1>0,πβ2=η2α2Γα2β2α2−1e−η2β2,α2,η2>0,where α1,η1,α2 and η2 are the hyperparameters. Then, the joint prior possibility distribution is
(35)πβ1,β2=πβ1πβ2=η1α1Γα1β1α1−1e−η1β1η2α2Γα2β2α2−1e−η2β2.The posterior probability distribution is calculated as follows:(36)πβ1,β2,γx,y=Lβ1,β2;x,yπβ1,β2∫0∞∫0∞Lβ1,β2;x,yπβ1,β2dβ1dβ2∝β1α1+r1−1e−β1η1+b1′β2α2+r2−1e−β2η2+b2′,where b1′ and b2′ were given in (4) and (5), respectively. For i=1,2, we can write
(37)βix,y~Gammaαi∗,ηi∗,where αi∗=αi+ri and ηi∗=ηi+bi′. [24] proposed α1=η1=α2=η2=0.001, and Robert [25] suggested an empirical Bayesian approach to determining the values of hyperparameters. According to the approach presented by Robert, to get α1 and η1, we maximize the following function:
(38)mxα1,η1=∫0∞fxα1,η1πβ1α1,η1dβ1=n1!η1α1eγ∑i=1r1xiΓα1n1−r1!∫0∞β1α1+r1−1e−β1η1+b1′dβ1=n1!η1α1Γα1+r1eγ∑i=1r1xiΓα1n1−r1!η1+b1′α1+r1.We have(39)Mxα1,η1=logmxα1,η1=logn1!n1−r1!+α1logη1+logΓα1+r1−logΓα1−α1+r1logη1+b1′+γ∑i=1r1xi.α1 and η1 are obtained by solving the following equations:
(40)∂M∂α1=logη1+ψr1+α1−ψα1−logη1+b1′=0,(41)∂M∂η1=α1η1−α1+r1η1+b1′=0,where ψ shows the digamma function. By solving Equation (41), η1 is
(42)η1=α1b1′r1.By substituting (42) into (40), we get
(43)0=logη1η1+b1′+ψr1+α1−ψα1=logα1r1+α1+ψr1+α1−ψα1=logα1r1+α1+∑k=0r1−11α1+k.Similarly, this method can be repeated to calculateα2 and η2. So, η2 is as follows:
(44)η2=α2b2′r2.Also,α2 is obtained by solving the following equation:
(45)0=logα2r2+α2+∑k=0r2−11α2+k.Abravesh et al. [4] showed that Equations (43) and (45) have no answer and set α1=α2=1 to solve this problem. Then, η1=b1′/r1 and η2=b2′/r2.


### 5.2. Posterior Distribution ofR

To calculate the posterior distribution ofR, we have the following transformations:
(46)R=β2β1+β2,(47)V=β2.Transformations (46) and (47) are equivalent to β1=V1−R/R and β2=V. The posterior distribution of R and V can be calculated by the following formula:
(48)πr,vx,y=J·πv1−rr,vx,y.In the above formula,J is called Jacobin and is calculated as follows:
(49)J=det∂β1∂v∂β2∂v∂β1∂r∂β2∂r=det1−rr1−vr20=vr2.The marginal distribution ofR is calculated from the joint distribution in (48) as
(50)πrx,y=∫0∞πIr,vx,ydu=∫0∞vr2·η1∗α1∗η2∗α2∗Γα1∗Γα2∗v1−rrα1∗−1e−η1∗v1−r/rvα2∗−1e−η2∗vdu=η1∗α1∗η2∗α2∗Γα1∗+α2∗Γα1∗Γα2∗·1/r−1α1∗−1r2η1∗1/r−1+η2∗α1∗+α2∗,0≤r≤1.


### 5.3. Jeffreys Noninformative Prior

In this section, by using Jeffreys noninformative prior [26], the Bayesian estimation of R is obtained. The Jeffreys prior is as follows:
(51)πJβ1,β2∝detIβ1,β2,where
(52)Iβ1,β2=−E∂2l∂β12E∂2l∂β1∂β2E∂2l∂β2∂β1E∂2l∂β22.Considering Jeffreys priorπJ∝1/β1β2, the marginal posterior distribution is given by
(53)βix,y~Gammari,bi′,i=1,2.Similar to the process in Subsection5.2, the marginal posterior distribution can be obtained:
(54)πJrx,y=b1′r1b2′r2Γr1+r2Γr1Γr2·1/r−1r1−1r2b1′1/r−1+b2′r1+r2,0≤r≤1.


### 5.4. Lindley Approximation

Lindley [27] proposed a method for approximating the ratio of integrals. The Lindley approximation of R can be calculated using the following formula:
(55)ERτdata=∫Rτelτ+Πτdτ∫elτ+Πτdτ≈Rτ^+12∑i,jρij+2ρiΠiσij+∑i,j,k,llijkρlσijσkl,where Σ=σijτ^=−lijτ∧−1, τ^ is MLE of τ and
(56)ρi=∂R∂τiτ=τ^,ρij=∂2R∂τi∂τjτ=τ^,Πi=∂Π∂τiτ=τ^,lijk=∂3l∂τi∂τj∂τkτ=τ^.Here,l is log-likelihood function and Π is log-prior distribution.


#### 5.4.1. Informative Prior

Based on the prior distribution (35) and R=β2/β1+β2, we obtain
(57)Π=logπβ1,β2=α1logη1−logΓα1+α1−1logβ1−η1β1+α2logη2−logΓα2+α2−1logβ2−η2β2.So,(58)Π1=∂Π∂β1=α1−1β1−η1,Π2=∂Π∂β2=α2−1β2−η2,ρ1=∂R∂β1=−β2β1+β22,ρ2=∂R∂β2=β1β1+β22,ρ11=∂2R∂β12=2β2β1+β23,ρ22=∂2R∂β22=−2β1β1+β23,ρ12=∂2R∂β2∂β1=β2−β1β1+β23.Inverse of the Hessian matrix is given by(59)Σ=β12r100β22r2.So,σ11=β12/r1,σ22=β22/r2,σ12=σ21=0 and
(60)liii=∂3l∂βi3=2riβi3,i=1,2.Finally, the Lindley approximation of the Bayes estimator ofR is
(61)RBayes≈Rτ^+12∑i,jρij+2ρiΠiσij+∑i,j,k,llijkρlσijσkl=Rβ^1,β^2+12∑i=12ρii+2ρiΠiσii+12∑i=12liiiρiσii2=Rβ^1,β^2+β^12r1β^2β∧1+β∧23+−β^2β∧1+β∧22α1−1β^1−η1+β^22r2−β^1β∧1+β∧23+β^1β∧1+β∧22α2−1β^2−η2+β^1β^2β∧1+β∧221r2−1r1.


#### 5.4.2. Noninformative Prior

Under Jeffreys prior (πJβ1,β2=1/β1β2), we have
(62)Π=−logβ1−logβ2,Π1=−1β1,Π2=−1β2.Therefore, the Bayes estimator ofR using the Lindley approximation is
(63)RBayes≈Rτ^+12∑i,jρij+2ρiΠiσij+∑i,j,k,llijkρlσijσkl=Rβ^1,β^2+12∑i=12ρii+2ρiΠiσii+12∑i=12liiiρiσii2=Rβ^1,β^2+β^12r1β^2β∧1+β∧23+β^2β^1β∧1+β∧22+β^22r2−β^1β∧1+β∧23+β^1β^2β∧1+β∧22+β^1β^2β∧1+β∧221r2−1r1.


### 5.5. Monte Carlo Integration

The Monte Carlo integration method was introduced by Metropolis and Ulam [28] and Neumann [29]. Let θ1,θ2,⋯,θk be a random sample from posterior density πθobservations. In this case, according to the strong law of large numbers, for large k, an approximation for the expected values of posterior is equal to
(64)Eζθobservation≃1k∑i=1kζθi.This method was very simple and does not involve complicated calculations. The only problem this method may have is generating a random sample of posterior density.Now, the Bayes estimator ofR is obtained using this method. Let βi1,βi2,⋯,βiN be the random sample from πβix,y,i=1,2, then for large N,
(65)ERx,y≈1N∑j=1NRβ1j,β2j=1N∑j=1Nβ2jβ1j+β2j.


#### 5.5.1. Informative Prior

To calculate the Bayes estimator ofR under the conjugate prior (35), we assume βi1,βi2,⋯,βiN~iidGammaαi+ri,ηi+bi′,i=1,2. So,
(66)R^MCI=1N∑j=1NRβ1j,β2j=1N∑j=1Nβ2jβ1j+β2j.


#### 5.5.2. Noninformative Prior

Under the Jeffreys prior, we considerβi1,βi2,⋯,βiN~iidGammari,bi′,i=1,2. Then,
(67)R^MCJ=1N∑j=1NRβ1j,β2j=1N∑j=1Nβ2jβ1j+β2j.


### 5.6. MCMC

To solve the stated problem of the Monte Carlo integration method, a more general method is used to generate approximate random variables from the posterior distribution, called the Markov chain Monte Carlo (MCMC) method [30]. The Metropolis-Hastings algorithm is used to create Markov chains with a given distribution. The application of this algorithm in mechanical physics was first developed by Metropolis et al. [30]. A few years later, Hastings and Keith generalized the algorithm in more statistical detail [31]. Using the MCMC method, the following integral is approximated:
(68)ERx,y=∫01rπrx,ydr.Letr1,r2,⋯,rl be an ergodic MCMC sample from πrx,y, we have
(69)ERx,y≈1l∑j=1lrj.


#### 5.6.1. Informative Prior

Considering the conjugate prior distribution (35) for β1 and β2, we obtain the posterior distribution (50). By generating an ergodic sample r1I,r2I,⋯,rlI from πI using the Metropolis-Hastings algorithm, the R Bayesian estimator is as follows:
(70)ERx,y≈1l∑j=1lrjI.


#### 5.6.2. Noninformative Prior

Similarly, using the Jeffreys prior and the posterior distribution (54) for β1 and β2, we generate an ergodic sample r1J,r2J,⋯,rlJ from πJ using the Metropolis-Hastings algorithm; the R Bayesian estimator is given by
(71)ERx,y≈1l∑j=1lrjJ.


## 5.1. Conjugate Informative Prior

Letβ1~Gammaα1,η1 and β2~Gammaα2,η2 and are independent. The PDF of these priors is as follows:
(34)πβ1=η1α1Γα1β1α1−1e−η1β1,α1,η1>0,πβ2=η2α2Γα2β2α2−1e−η2β2,α2,η2>0,where α1,η1,α2 and η2 are the hyperparameters. Then, the joint prior possibility distribution is
(35)πβ1,β2=πβ1πβ2=η1α1Γα1β1α1−1e−η1β1η2α2Γα2β2α2−1e−η2β2.The posterior probability distribution is calculated as follows:(36)πβ1,β2,γx,y=Lβ1,β2;x,yπβ1,β2∫0∞∫0∞Lβ1,β2;x,yπβ1,β2dβ1dβ2∝β1α1+r1−1e−β1η1+b1′β2α2+r2−1e−β2η2+b2′,where b1′ and b2′ were given in (4) and (5), respectively. For i=1,2, we can write
(37)βix,y~Gammaαi∗,ηi∗,where αi∗=αi+ri and ηi∗=ηi+bi′. [24] proposed α1=η1=α2=η2=0.001, and Robert [25] suggested an empirical Bayesian approach to determining the values of hyperparameters. According to the approach presented by Robert, to get α1 and η1, we maximize the following function:
(38)mxα1,η1=∫0∞fxα1,η1πβ1α1,η1dβ1=n1!η1α1eγ∑i=1r1xiΓα1n1−r1!∫0∞β1α1+r1−1e−β1η1+b1′dβ1=n1!η1α1Γα1+r1eγ∑i=1r1xiΓα1n1−r1!η1+b1′α1+r1.We have(39)Mxα1,η1=logmxα1,η1=logn1!n1−r1!+α1logη1+logΓα1+r1−logΓα1−α1+r1logη1+b1′+γ∑i=1r1xi.α1 and η1 are obtained by solving the following equations:
(40)∂M∂α1=logη1+ψr1+α1−ψα1−logη1+b1′=0,(41)∂M∂η1=α1η1−α1+r1η1+b1′=0,where ψ shows the digamma function. By solving Equation (41), η1 is
(42)η1=α1b1′r1.By substituting (42) into (40), we get
(43)0=logη1η1+b1′+ψr1+α1−ψα1=logα1r1+α1+ψr1+α1−ψα1=logα1r1+α1+∑k=0r1−11α1+k.Similarly, this method can be repeated to calculateα2 and η2. So, η2 is as follows:
(44)η2=α2b2′r2.Also,α2 is obtained by solving the following equation:
(45)0=logα2r2+α2+∑k=0r2−11α2+k.Abravesh et al. [4] showed that Equations (43) and (45) have no answer and set α1=α2=1 to solve this problem. Then, η1=b1′/r1 and η2=b2′/r2.


## 5.2. Posterior Distribution ofR

To calculate the posterior distribution ofR, we have the following transformations:
(46)R=β2β1+β2,(47)V=β2.Transformations (46) and (47) are equivalent to β1=V1−R/R and β2=V. The posterior distribution of R and V can be calculated by the following formula:
(48)πr,vx,y=J·πv1−rr,vx,y.In the above formula,J is called Jacobin and is calculated as follows:
(49)J=det∂β1∂v∂β2∂v∂β1∂r∂β2∂r=det1−rr1−vr20=vr2.The marginal distribution ofR is calculated from the joint distribution in (48) as
(50)πrx,y=∫0∞πIr,vx,ydu=∫0∞vr2·η1∗α1∗η2∗α2∗Γα1∗Γα2∗v1−rrα1∗−1e−η1∗v1−r/rvα2∗−1e−η2∗vdu=η1∗α1∗η2∗α2∗Γα1∗+α2∗Γα1∗Γα2∗·1/r−1α1∗−1r2η1∗1/r−1+η2∗α1∗+α2∗,0≤r≤1.


## 5.3. Jeffreys Noninformative Prior

In this section, by using Jeffreys noninformative prior [26], the Bayesian estimation of R is obtained. The Jeffreys prior is as follows:
(51)πJβ1,β2∝detIβ1,β2,where
(52)Iβ1,β2=−E∂2l∂β12E∂2l∂β1∂β2E∂2l∂β2∂β1E∂2l∂β22.Considering Jeffreys priorπJ∝1/β1β2, the marginal posterior distribution is given by
(53)βix,y~Gammari,bi′,i=1,2.Similar to the process in Subsection5.2, the marginal posterior distribution can be obtained:
(54)πJrx,y=b1′r1b2′r2Γr1+r2Γr1Γr2·1/r−1r1−1r2b1′1/r−1+b2′r1+r2,0≤r≤1.


## 5.4. Lindley Approximation

Lindley [27] proposed a method for approximating the ratio of integrals. The Lindley approximation of R can be calculated using the following formula:
(55)ERτdata=∫Rτelτ+Πτdτ∫elτ+Πτdτ≈Rτ^+12∑i,jρij+2ρiΠiσij+∑i,j,k,llijkρlσijσkl,where Σ=σijτ^=−lijτ∧−1, τ^ is MLE of τ and
(56)ρi=∂R∂τiτ=τ^,ρij=∂2R∂τi∂τjτ=τ^,Πi=∂Π∂τiτ=τ^,lijk=∂3l∂τi∂τj∂τkτ=τ^.Here,l is log-likelihood function and Π is log-prior distribution.


### 5.4.1. Informative Prior

Based on the prior distribution (35) and R=β2/β1+β2, we obtain
(57)Π=logπβ1,β2=α1logη1−logΓα1+α1−1logβ1−η1β1+α2logη2−logΓα2+α2−1logβ2−η2β2.So,(58)Π1=∂Π∂β1=α1−1β1−η1,Π2=∂Π∂β2=α2−1β2−η2,ρ1=∂R∂β1=−β2β1+β22,ρ2=∂R∂β2=β1β1+β22,ρ11=∂2R∂β12=2β2β1+β23,ρ22=∂2R∂β22=−2β1β1+β23,ρ12=∂2R∂β2∂β1=β2−β1β1+β23.Inverse of the Hessian matrix is given by(59)Σ=β12r100β22r2.So,σ11=β12/r1,σ22=β22/r2,σ12=σ21=0 and
(60)liii=∂3l∂βi3=2riβi3,i=1,2.Finally, the Lindley approximation of the Bayes estimator ofR is
(61)RBayes≈Rτ^+12∑i,jρij+2ρiΠiσij+∑i,j,k,llijkρlσijσkl=Rβ^1,β^2+12∑i=12ρii+2ρiΠiσii+12∑i=12liiiρiσii2=Rβ^1,β^2+β^12r1β^2β∧1+β∧23+−β^2β∧1+β∧22α1−1β^1−η1+β^22r2−β^1β∧1+β∧23+β^1β∧1+β∧22α2−1β^2−η2+β^1β^2β∧1+β∧221r2−1r1.


### 5.4.2. Noninformative Prior

Under Jeffreys prior (πJβ1,β2=1/β1β2), we have
(62)Π=−logβ1−logβ2,Π1=−1β1,Π2=−1β2.Therefore, the Bayes estimator ofR using the Lindley approximation is
(63)RBayes≈Rτ^+12∑i,jρij+2ρiΠiσij+∑i,j,k,llijkρlσijσkl=Rβ^1,β^2+12∑i=12ρii+2ρiΠiσii+12∑i=12liiiρiσii2=Rβ^1,β^2+β^12r1β^2β∧1+β∧23+β^2β^1β∧1+β∧22+β^22r2−β^1β∧1+β∧23+β^1β^2β∧1+β∧22+β^1β^2β∧1+β∧221r2−1r1.


## 5.4.1. Informative Prior

Based on the prior distribution (35) and R=β2/β1+β2, we obtain
(57)Π=logπβ1,β2=α1logη1−logΓα1+α1−1logβ1−η1β1+α2logη2−logΓα2+α2−1logβ2−η2β2.So,(58)Π1=∂Π∂β1=α1−1β1−η1,Π2=∂Π∂β2=α2−1β2−η2,ρ1=∂R∂β1=−β2β1+β22,ρ2=∂R∂β2=β1β1+β22,ρ11=∂2R∂β12=2β2β1+β23,ρ22=∂2R∂β22=−2β1β1+β23,ρ12=∂2R∂β2∂β1=β2−β1β1+β23.Inverse of the Hessian matrix is given by(59)Σ=β12r100β22r2.So,σ11=β12/r1,σ22=β22/r2,σ12=σ21=0 and
(60)liii=∂3l∂βi3=2riβi3,i=1,2.Finally, the Lindley approximation of the Bayes estimator ofR is
(61)RBayes≈Rτ^+12∑i,jρij+2ρiΠiσij+∑i,j,k,llijkρlσijσkl=Rβ^1,β^2+12∑i=12ρii+2ρiΠiσii+12∑i=12liiiρiσii2=Rβ^1,β^2+β^12r1β^2β∧1+β∧23+−β^2β∧1+β∧22α1−1β^1−η1+β^22r2−β^1β∧1+β∧23+β^1β∧1+β∧22α2−1β^2−η2+β^1β^2β∧1+β∧221r2−1r1.


## 5.4.2. Noninformative Prior

Under Jeffreys prior (πJβ1,β2=1/β1β2), we have
(62)Π=−logβ1−logβ2,Π1=−1β1,Π2=−1β2.Therefore, the Bayes estimator ofR using the Lindley approximation is
(63)RBayes≈Rτ^+12∑i,jρij+2ρiΠiσij+∑i,j,k,llijkρlσijσkl=Rβ^1,β^2+12∑i=12ρii+2ρiΠiσii+12∑i=12liiiρiσii2=Rβ^1,β^2+β^12r1β^2β∧1+β∧23+β^2β^1β∧1+β∧22+β^22r2−β^1β∧1+β∧23+β^1β^2β∧1+β∧22+β^1β^2β∧1+β∧221r2−1r1.


## 5.5. Monte Carlo Integration

The Monte Carlo integration method was introduced by Metropolis and Ulam [28] and Neumann [29]. Let θ1,θ2,⋯,θk be a random sample from posterior density πθobservations. In this case, according to the strong law of large numbers, for large k, an approximation for the expected values of posterior is equal to
(64)Eζθobservation≃1k∑i=1kζθi.This method was very simple and does not involve complicated calculations. The only problem this method may have is generating a random sample of posterior density.Now, the Bayes estimator ofR is obtained using this method. Let βi1,βi2,⋯,βiN be the random sample from πβix,y,i=1,2, then for large N,
(65)ERx,y≈1N∑j=1NRβ1j,β2j=1N∑j=1Nβ2jβ1j+β2j.


### 5.5.1. Informative Prior

To calculate the Bayes estimator ofR under the conjugate prior (35), we assume βi1,βi2,⋯,βiN~iidGammaαi+ri,ηi+bi′,i=1,2. So,
(66)R^MCI=1N∑j=1NRβ1j,β2j=1N∑j=1Nβ2jβ1j+β2j.


### 5.5.2. Noninformative Prior

Under the Jeffreys prior, we considerβi1,βi2,⋯,βiN~iidGammari,bi′,i=1,2. Then,
(67)R^MCJ=1N∑j=1NRβ1j,β2j=1N∑j=1Nβ2jβ1j+β2j.


## 5.5.1. Informative Prior

To calculate the Bayes estimator ofR under the conjugate prior (35), we assume βi1,βi2,⋯,βiN~iidGammaαi+ri,ηi+bi′,i=1,2. So,
(66)R^MCI=1N∑j=1NRβ1j,β2j=1N∑j=1Nβ2jβ1j+β2j.


## 5.5.2. Noninformative Prior

Under the Jeffreys prior, we considerβi1,βi2,⋯,βiN~iidGammari,bi′,i=1,2. Then,
(67)R^MCJ=1N∑j=1NRβ1j,β2j=1N∑j=1Nβ2jβ1j+β2j.


## 5.6. MCMC

To solve the stated problem of the Monte Carlo integration method, a more general method is used to generate approximate random variables from the posterior distribution, called the Markov chain Monte Carlo (MCMC) method [30]. The Metropolis-Hastings algorithm is used to create Markov chains with a given distribution. The application of this algorithm in mechanical physics was first developed by Metropolis et al. [30]. A few years later, Hastings and Keith generalized the algorithm in more statistical detail [31]. Using the MCMC method, the following integral is approximated:
(68)ERx,y=∫01rπrx,ydr.Letr1,r2,⋯,rl be an ergodic MCMC sample from πrx,y, we have
(69)ERx,y≈1l∑j=1lrj.


### 5.6.1. Informative Prior

Considering the conjugate prior distribution (35) for β1 and β2, we obtain the posterior distribution (50). By generating an ergodic sample r1I,r2I,⋯,rlI from πI using the Metropolis-Hastings algorithm, the R Bayesian estimator is as follows:
(70)ERx,y≈1l∑j=1lrjI.


### 5.6.2. Noninformative Prior

Similarly, using the Jeffreys prior and the posterior distribution (54) for β1 and β2, we generate an ergodic sample r1J,r2J,⋯,rlJ from πJ using the Metropolis-Hastings algorithm; the R Bayesian estimator is given by
(71)ERx,y≈1l∑j=1lrjJ.


## 5.6.1. Informative Prior

Considering the conjugate prior distribution (35) for β1 and β2, we obtain the posterior distribution (50). By generating an ergodic sample r1I,r2I,⋯,rlI from πI using the Metropolis-Hastings algorithm, the R Bayesian estimator is as follows:
(70)ERx,y≈1l∑j=1lrjI.


## 5.6.2. Noninformative Prior

Similarly, using the Jeffreys prior and the posterior distribution (54) for β1 and β2, we generate an ergodic sample r1J,r2J,⋯,rlJ from πJ using the Metropolis-Hastings algorithm; the R Bayesian estimator is given by
(71)ERx,y≈1l∑j=1lrjJ.


## 6. Bayesian Credible Interval

In fact, the Bayesian view offers confidence interval that are more realistic than its classic counterpart. We start this section with two definitions.Definition 3.
SetCx is called a ξ-credible region whenever
(72)Pπθ∈Cxx≥1−ξ,where Pπ is the θ posterior probability function of condition x.Definition 4.
Theξ-credible region Cx is called a region with the highest posterior density (HPD) whenever it can be written as follows:
(73)Cxν=θ:πθx≥ν,where ν is the largest fixed number that applies to
(74)Pπθ∈Cxν≥1−ξ.Although the1001−ξ% HPD interval is an optimal answer among the ξ-credible intervals, in some cases, it is not easy to calculate directly, and approximate methods must be used to obtain it [32]. It is usually easier to calculate approximate intervals with equal tails than HPD interval [33]. Chen and Shao [34] have proposed an algorithm to construct an approximate HPD interval. We obtain confidence intervals of equal tails and HPD.


### 6.1. An Equi-Tailed Bayesian Credible Interval

In this subsection, confidence intervals with equal tails are calculated under the conjugate and Jeffreys prior distributions.


#### 6.1.1. Informative Prior

We use the prior distribution (35). The posterior distributions of β1 and β2 are as follows:
(75)βix,y~indGammaαi∗,ηi∗,i=1,2.Therefore, one can conclude(76)2ηi∗βix,y~χ22αi∗,∀i=1,2.Given that the posterior distributions ofβ1 and β2 are independent, we get
(77)η1∗α2∗α1∗η2∗·β1x,yβ2x,y~F2α1∗,2α2∗.Thus, a1001−ξ% Bayesian credible interval with equal tails for R under the conjugate prior is
(78)P11+α1∗η2∗/η1∗α2∗F1−ξ/22α1∗,2α2∗<R<11+α1∗η2∗/η1∗α2∗Fξ/22α1∗,2α2∗=1−ξ.


#### 6.1.2. Noninformative Prior

Similarly, considering the Jeffreys prior, the posterior distributions ofβ1 and β2 are β1x,y~indGammar1,b1′ and β2x,y~indGammar2,b2′, respectively. So, we have
(79)2bi′βix,y~χ22ri,i=1,2.Hence,(80)b1′r2r1b2′·β1x,yβ2x,y~F2r1,2r2.A1001−ξ% Bayesian credible interval with equal tails for R under the Jeffreys prior is
(81)P11+r1b2′/b1′r2F1−ξ/22r1,2r2<R<11+r1b2′/b1′r2Fξ/22r1,2r2=1−ξ.


### 6.2. HPD Interval

As mentioned earlier, it is difficult to obtain the HPD interval directly. Therefore, in this subsection, the Chen-Shao algorithm [34] is used to calculate the approximate HPD interval for R. This algorithm is expressed as follows.Algorithm 1: Chen-Shao algorithm for R.
1: Generate a MCMC sampleRi,i=1,2,⋯,k from πRx,y;2: SortRi,i=1,2,⋯,l and suppose R1≤R2≤⋯≤Rl;3: ConsiderCj=Rj,Rj+1−ξl,j=1,2,⋯,l−1−ξl;4: ConsiderWj=Rj+1−ξl−Rj;5: Selectj′ such that Wj′=minWj,j=1,2,⋯,l−1−ξl;6: IntroduceCj′ as a 1001−ξ% HPD interval for R.In Step 3,Cj’s are 1001−ξ% credible intervals for R. To obtain the HPD interval under the conjugate and Jeffreys priors, it is enough to substitute the posterior distributions (50) and (54) in Step 1 of Algorithm 1.


## 6.1. An Equi-Tailed Bayesian Credible Interval

In this subsection, confidence intervals with equal tails are calculated under the conjugate and Jeffreys prior distributions.


### 6.1.1. Informative Prior

We use the prior distribution (35). The posterior distributions of β1 and β2 are as follows:
(75)βix,y~indGammaαi∗,ηi∗,i=1,2.Therefore, one can conclude(76)2ηi∗βix,y~χ22αi∗,∀i=1,2.Given that the posterior distributions ofβ1 and β2 are independent, we get
(77)η1∗α2∗α1∗η2∗·β1x,yβ2x,y~F2α1∗,2α2∗.Thus, a1001−ξ% Bayesian credible interval with equal tails for R under the conjugate prior is
(78)P11+α1∗η2∗/η1∗α2∗F1−ξ/22α1∗,2α2∗<R<11+α1∗η2∗/η1∗α2∗Fξ/22α1∗,2α2∗=1−ξ.


### 6.1.2. Noninformative Prior

Similarly, considering the Jeffreys prior, the posterior distributions ofβ1 and β2 are β1x,y~indGammar1,b1′ and β2x,y~indGammar2,b2′, respectively. So, we have
(79)2bi′βix,y~χ22ri,i=1,2.Hence,(80)b1′r2r1b2′·β1x,yβ2x,y~F2r1,2r2.A1001−ξ% Bayesian credible interval with equal tails for R under the Jeffreys prior is
(81)P11+r1b2′/b1′r2F1−ξ/22r1,2r2<R<11+r1b2′/b1′r2Fξ/22r1,2r2=1−ξ.


## 6.1.1. Informative Prior

We use the prior distribution (35). The posterior distributions of β1 and β2 are as follows:
(75)βix,y~indGammaαi∗,ηi∗,i=1,2.Therefore, one can conclude(76)2ηi∗βix,y~χ22αi∗,∀i=1,2.Given that the posterior distributions ofβ1 and β2 are independent, we get
(77)η1∗α2∗α1∗η2∗·β1x,yβ2x,y~F2α1∗,2α2∗.Thus, a1001−ξ% Bayesian credible interval with equal tails for R under the conjugate prior is
(78)P11+α1∗η2∗/η1∗α2∗F1−ξ/22α1∗,2α2∗<R<11+α1∗η2∗/η1∗α2∗Fξ/22α1∗,2α2∗=1−ξ.


## 6.1.2. Noninformative Prior

Similarly, considering the Jeffreys prior, the posterior distributions ofβ1 and β2 are β1x,y~indGammar1,b1′ and β2x,y~indGammar2,b2′, respectively. So, we have
(79)2bi′βix,y~χ22ri,i=1,2.Hence,(80)b1′r2r1b2′·β1x,yβ2x,y~F2r1,2r2.A1001−ξ% Bayesian credible interval with equal tails for R under the Jeffreys prior is
(81)P11+r1b2′/b1′r2F1−ξ/22r1,2r2<R<11+r1b2′/b1′r2Fξ/22r1,2r2=1−ξ.


## 6.2. HPD Interval

As mentioned earlier, it is difficult to obtain the HPD interval directly. Therefore, in this subsection, the Chen-Shao algorithm [34] is used to calculate the approximate HPD interval for R. This algorithm is expressed as follows.Algorithm 1: Chen-Shao algorithm for R.
1: Generate a MCMC sampleRi,i=1,2,⋯,k from πRx,y;2: SortRi,i=1,2,⋯,l and suppose R1≤R2≤⋯≤Rl;3: ConsiderCj=Rj,Rj+1−ξl,j=1,2,⋯,l−1−ξl;4: ConsiderWj=Rj+1−ξl−Rj;5: Selectj′ such that Wj′=minWj,j=1,2,⋯,l−1−ξl;6: IntroduceCj′ as a 1001−ξ% HPD interval for R.In Step 3,Cj’s are 1001−ξ% credible intervals for R. To obtain the HPD interval under the conjugate and Jeffreys priors, it is enough to substitute the posterior distributions (50) and (54) in Step 1 of Algorithm 1.


## 7. Simulation Study

In this section, we use simulation to compare estimators and confidence intervals of R. Therefore, for different sample sizes, different numbers of type II censorship, and differentR values with 1000 repetitions, bias and mean square error (MSE) values of R estimators are calculated. For the conjugate informative prior distribution, hyperparameters α1=α2=1,η1=b1′/r1 and η2=b2′/r2 are considered. Tables 1–3 show biases and MSEs of point estimators with R=1/2,R=2/3, and R=1/3, respectively. Using Algorithm 2, type II censored samples are generated from two independent Gompertz distributions.Algorithm 2: The type II censored sample generation algorithm.
1: Selectn1,n2,β1,β2,r1,r2, and γ values;2: Generateτ1,τ2,⋯,τn1 and δ1,δ2,⋯,δn2 from the SED;3: Considerxi=1/γlogγ/β1τi+1,i=1,2,⋯,n1 and yj=1/γlogγ/β2δj+1,i=1,2,⋯,n2;4: Sortxis and yjs and suppose x1<x2<⋯<xn1 and y1<y2<⋯<yn2;5: Reportx1,x2,⋯,xr1 and y1,y2,⋯,yr2 as the type II censored samples from Gompertzβ1,γ and Gompertzβ2,γ, respectively.Table 1
The bias and MSE values (MSE in parentheses) of estimators forβ1=1,β2=1,γ=1,R=1/2.
n1n2r1r2PriorMLELindleyMCMCMC10101010Conjugate0.011339 (0.015528)0.010107 (0.012152)0.010295 (0.012573)0.006669 (0.006495)Jeffreys—0.010822 (0.014271)0.011231 (0.014295)0.007375 (0.007236)98Conjugate0.009474 (0.021751)0.003927 (0.016272)0.007728 (0.016904)0.005395 (0.008246)Jeffreys—0.004289 (0.019732)0.007793 (0.019736)0.004971 (0.009175)88Conjugate-0.000753 (0.021541)-0.000852 (0.016715)-0.000681 (0.017157)-0.000596 (0.008737)Jeffreys—-0.000815 (0.019670)-0.000749 (0.019676)-0.000502 (0.008889)10301030Conjugate-0.015306 (0.010153)0.010855 (0.008091)-0.006733 (0.008151)-0.005904 (0.004985)Jeffreys—0.009125 (0.009500)-0.006959 (0.009400)-0.006247 (0.005666)928Conjugate-0.003033 (0.009196)0.024440 (0.007826)0.004501 (0.007476)0.002187 (0.004340)Jeffreys—0.024301 (0.009185)0.006082 (0.008662)0.003254 (0.004896)827Conjugate-0.017882 (0.016928)0.015887 (0.013174)-0.006873 (0.013339)-0.005408 (0.007667)Jeffreys—0.013523 (0.015708)-0.007212 (0.015539)-0.006585 (0.008730)30303030Conjugate0.007389 (0.004586)0.007087 (0.004188)0.007466 (0.004211)0.006229 (0.003175)Jeffreys—0.007269 (0.004444)0.007113 (0.004477)0.006013 (0.003299)2927Conjugate0.000869 (0.004186)-0.000089 (0.003779)0.000583 (0.003835)0.001015 (0.002807)Jeffreys—-0.000090 (0.004044)0.000408 (0.004046)0.001044 (0.002948)2727Conjugate-0.002992 (0.005576)-0.002802 (0.005017)-0.002798 (0.005059)-0.002345 (0.003726)Jeffreys—-0.002939 (0.005384)-0.002854 (0.005423)-0.002606 (0.003900)Table 2
The bias and MSE values (MSE in parentheses) of estimators forβ1=1,β2=2,γ=1,R=2/3.
n1n2r1r2PriorMLELindleyMCMCMC10101010Conjugate0.009500 (0.012304)-0.015630 (0.010022)-0.012485 (0.010080)-0.053973 (0.008464)Jeffreys—0.002976 (0.011512)0.003521 (0.011480)-0.045037 (0.008308)98Conjugate0.047141 (0.019723)0.013442 (0.015408)0.019974 (0.015494)-0.030126 (0.010706)Jeffreys—0.035412 (0.018268)0.037776 (0.018229)-0.019705 (0.011154)88Conjugate0.002840 (0.019525)-0.023443 (0.015882)-0.020104 (0.016047)-0.062010 (0.013115)Jeffreys—-0.004222 (0.018351)-0.004391 (0.018341)-0.053220 (0.013700)10301030Conjugate-0.002736 (0.008137)0.008917 (0.006132)-0.005916 (0.006567)-0.038466 (0.006042)Jeffreys—0.014354 (0.007555)-0.000261 (0.007695)-0.034790 (0.006342)928Conjugate-0.000636 (0.011272)0.013486 (0.008512)-0.003448 (0.009072)-0.037787 (0.007786)Jeffreys—0.018633 (0.010370)0.002493 (0.010620)-0.036707 (0.008483)827Conjugate-0.011756 (0.007082)0.006556 (0.004932)-0.013705 (0.005678)-0.049995 (0.006245)Jeffreys—0.012011 (0.006199)-0.007743 (0.006572)-0.046598 (0.006817)30303030Conjugate0.003700 (0.003215)-0.005787 (0.002979)-0.005248 (0.002967)-0.025586 (0.003080)Jeffreys—0.001298 (0.003137)0.001252 (0.003141)-0.019025 (0.002924)2927Conjugate0.009701 (0.009701)-0.001228 (0.004512)-0.000352 (0.004518)-0.021084 (0.004090)Jeffreys—0.006303 (0.004813)0.006694 (0.004831)-0.014817 (0.004119)2727Conjugate0.013939 (0.004330)0.003420 (0.003420)0.004086 (0.003858)-0.018673 (0.003403)Jeffreys—0.011203 (0.004182)0.011117 (0.004171)-0.013123 (0.003508)Table 3
The bias and MSE values (MSE in parentheses) of estimators forβ1=2,β2=1,γ=1,R=1/3.
n1n2r1r2PriorMLELindleyMCMCMC10101010Conjugate-0.006236 (0.013546)0.017747 (0.011277)0.014489 (0.011353)0.055189 (0.009427)Jeffreys—0.000054 (0.012776)-0.000131 (0.012800)0.046938 (0.009554)98Conjugate-0.020955 (0.020210)0.001998 (0.015037)0.002221 (0.015536)0.049048 (0.010972)Jeffreys—-0.017516 (0.018642)-0.014752 (0.018694)0.039356 (0.011794)88Conjugate-0.036934 (0.018266)-0.006422 (0.014205)-0.010386 (0.014360)0.039886 (0.010667)Jeffreys—-0.028748 (0.016873)-0.028721 (0.016784)0.027892 (0.011565)10301030Conjugate-0.019006 (0.006476)0.029157 (0.006079)0.010961 (0.005102)0.044632 (0.005154)Jeffreys—0.006823 (0.006389)-0.007084 (0.006079)0.031843 (0.004776)928Conjugate-0.022646 (0.008856)0.029192 (0.008100)0.008836 (0.006968)0.044723 (0.006398)Jeffreys—0.005889 (0.008797)-0.009568 (0.008326)0.031616 (0.006042)827Conjugate-0.020403 (0.009430)0.040819 (0.009588)0.016503 (0.007684)0.053626 (0.007476)Jeffreys—0.012477 (0.009715)-0.005752 (0.008934)0.035300 (0.007476)30303030Conjugate-0.005453 (0.004098)0.003841 (0.003746)0.003399 (0.003739)0.023368 (0.003607)Jeffreys—-0.003058 (0.003991)-0.002791 (0.004009)0.018217 (0.003467)2927Conjugate-0.003748 (0.004346)0.005040 (0.004015)0.005143 (0.004039)0.026241 (0.003894)Jeffreys—-0.002042 (0.004232)-0.001604 (0.004234)0.019994 (0.003906)2727Conjugate-0.007624 (0.003878)0.002911 (0.003491)0.002431 (0.003503)0.023633 (0.003301)Jeffreys—-0.004931 (0.003755)-0.005036 (0.003742)0.017996 (0.003314)The results of the proposed methods for point estimation are summarized in Tables1–3. Based on these tables, the following results can be achieved:
(i)
The MCMC method has the lowest MSE(ii)
For small sample size, the Bayesian method performs better than the MLE method(iii)
The MSE of Bayesian estimators under conjugate and Jeffreys priors is not significantly different(iv)
The MSE of all estimators decreases significantly with increasing sample sizeTables4–6 compare the proposed confidence intervals using the interval lengths (L) and coverage probabilities (CPs). From these tables, the following results are obtained:
(i)
The Bayesian credible intervals equi-tailed under the conjugate priors have the shortest, and the exact interval has the longest interval length(ii)
For classical and Bayesian methods, theL’s and the CPs have been improved by increasing the sample size.Table 4
L and CP of confidence interval for R=1/2,β1=1,β2=1 and γ=1.
n1n2r1r2AsymptoticExactCredible equi-tailedConjugateCredible equi-tailedJeffreysHPDConjugateHPDJeffreys10101010LCP0.4106230.4179160.3280470.3378850.3853560.4000560.9950.9000.9360.9390.9100.90098LCP0.4354110.4471570.3427570.3540500.4064170.4236520.9870.8750.9220.9200.8860.87088LCP0.4430060.4542250.3473410.3587960.4128290.4310750.7250.8570.9220.9160.8690.85510301030LCP0.3448990.3527220.2878690.2955430.3274920.3388300.2420.9290.9540.9530.9320.922928LCP0.3597450.3696850.2977460.3063820.3406500.3536031.0000.9320.9650.9550.9440.931827LCP0.3739870.3864980.3066790.3165320.3531560.3677651.0000.9120.9500.9420.9260.91030303030LCP0.2484070.2496670.2232030.2267090.2420770.2457371.0000.9340.9460.9430.9430.9402927LCP0.2572440.2586820.2300900.2332500.2502270.2542821.0000.9410.9490.9490.9420.9382727LCP0.2610870.2626380.2333560.2363270.2538000.2580141.0000.9280.9370.9350.9290.927Table 5
L and CP of confidence interval for R=2/3,β1=1,β2=2 and γ=1.
n1n2r1r2AsymptoticExactCredible equi-tailedConjugateCredible equi-tailedJeffreysHPDConjugateHPDJeffreys10101010LCP0.3581330.3714380.3111950.3174410.3495150.3550550.9870.8670.9070.9080.8820.86998LCP0.3707740.3905390.3244410.3320620.3651060.3702520.9250.8500.9020.9020.8690.84488LCP0.3830270.4032750.3299600.3383160.3744780.3807630.7240.8500.8940.9000.8740.84910301030LCP0.3041990.3076650.2603160.2648830.2900420.2969470.9960.8860.9070.8970.9050.883928LCP0.3224090.3266110.2717760.2772040.3056270.3138881.0000.9090.9080.9060.9220.911827LCP0.3333310.3380900.2777640.2835950.3144200.3236530.7590.8770.8860.8790.8990.88030303030LCP0.2193600.2220870.2048150.2065640.2172820.2184711.0000.9300.9420.9380.9380.9272927LCP0.2252290.2282340.2106350.2119620.2233880.2245141.0000.9030.9150.9180.9230.9062727LCP0.2297740.2329500.2141640.2154510.2274830.2288001.0000.9150.9200.9190.9280.920Table 6
L and CP of confidence interval for R=1/3,β1=2,β2=1 and γ=1.
n1n2r1r2AsymptoticExactCredible equi-tailedConjugateCredible equi-tailedJeffreysHPDConjugateHPDJeffreys10101010LCP0.3610560.3750950.3121000.3188560.3520940.3577470.9560.9160.9300.9360.9250.90098LCP0.3810430.3962600.3245650.3322430.3694550.3763890.9390.8570.9060.8910.8800.85888LCP0.3819050.4006020.3297600.3371890.3738950.3796860.9220.8450.8990.9010.8760.84710301030LCP0.3012830.3204920.2803370.2862960.3027040.3061930.8230.9270.9320.9480.9480.930928LCP0.3153210.3376850.2908390.2979120.3169070.3210670.0950.9330.9340.9380.9360.919827LCP0.3261390.3535990.3017780.3093640.3294060.3340211.0000.9230.9420.9510.9390.92930303030LCP0.2198060.2224440.2058920.2066720.2176760.2188901.0000.9310.9250.9260.9350.9252927LCP0.2278590.2307040.2116580.2134100.2251770.2266501.0000.9240.9360.9390.9320.9212727LCP0.2296010.2327000.2140870.2152310.2273140.2286321.0000.9090.9190.9190.9180.906Among these interval estimators, the CPs of the exact intervals are close to nominal level95%.
(iii)
The CPs of the Bayesian credible intervals equi-tailed are almost the same under the conjugate and Jeffreys priors(iv)
WhenR is close to 1/2, the HPD intervals have an overestimate in estimating the CP(v)
WhenR is far from 1/2, the HPD intervals have an underestimate in estimating the CP


## 8. Application

The sample lifetime of a steel particular type under two different pressures of 35.5X and 35 Y is reported in Table 7. This data contains 20 observations in each sample. This data has been studied by Kimber [35]. To verify that the data have a Gompertz distribution, we perform the Kolmogorov-Smirnov test. Based on the test statistics and P value, it is concluded that X has a Gompertz distribution with parameters β1=0.0013 and γ=0.0027D=0.2,P‐value=0.832 and Y has a Gompertz distribution with parameters β2=0.00093 and γ=0.0027D=0.35,P‐value=0.1745. We consider r1=r2=2. In Bayesian estimation, the values of the hyperparameters are α1=α2=1,η1=b1′/r1 and η2=b2′/r2. Now, we apply the proposed methods for point estimation and confidence interval estimation to this data. The results are summarized in Tables 8 and 9. Based on the value of R=PX>Y=0.417, it can be concluded that the lifetime of steel under pressure 35 is greater than the lifetime of steel under pressure 35.5.Table 7
Real data.
X156442285247173168253112125286166202852261133365559227309702Y2305681101218169115285342178280734431271305177143129326493381Table 8
Estimators ofR for real data.
PriorMLELindleyMCMCMCConjugate0.41699460.41924170.42024600.4294883Jeffreys—0.41923670.41588640.4422381Table 9
Confidence intervals ofR for real data.
AsymptoticExactCredible equi-tailedConjugateCredible equi-tailedJeffreysHPDConjugateHPDJeffreysLower0.25816560.24146670.27276560.26915750.30789940.2989704Upper0.57582360.55543830.57699030.58143210.58044700.5704925L0.3176580.3139720.2725480.2715220.3042250.312275


## 9. Conclusion

This paper proposed a classical and Bayesian inference for stress-strength reliability of Gompertz distribution under type II censoring. First, the MLE ofR was obtained. Then, exact and asymptotic confidence intervals for R were presented. In addition, Bayesian estimators of R obtained using Lindley approximation, Monte Carlo, and MCMC under conjugate informative and Jeffreys noninformative priors were discussed. Also, Bayesian credible intervals with equi-tailed and HPD intervals under conjugate and Jeffreys prior distributions were obtained. The proposed methods were compared with simulation studies. Finally, the application of these methods was examined with a real data.



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