# Partially Accelerated Model for Analyzing Competing Risks Data from Gompertz Population under Type-I Generalized Hybrid Censoring Scheme

Authors: Abdulaziz S. Alghamdi
Journal: Complexity
 (2021)

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## Abstract

In reliability engineering and lifetime analysis, many units of the product fail with different causes of failure, and some tests require stress higher than normal stress. Also, we need to design the life experiments which present methodology for formulating scientific and engineering problems using statistical models. So, in this paper, we adopted a partially constant stress accelerated life test model to present times to failure in a small period of time for Gompertz life products. Also, considering that, units are failing with the only two independent causes of failure and tested under type-I generalized hybrid censoring scheme the data built. Obtained data are analyzed with two methods of estimations, maximum likelihood and Bayes methods. These two methods are used to construct the point and interval estimators with the help of the MCMC method. The developed results are measured and compared under Monte Carlo studying. Also, a data set is analyzed for illustration purposes. Finally, some comments are presented to describe the numerical results.


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## Body

## 1. Introduction

The data under life-testing experiments may be complete or censoring data; the words of complete data set are used when the time to failure for all units under the test is obtained. But, the word censoring data is used when some but not all data about tested units is obtained. Type-I and type-II censoring schemes are from the oldest censoring schemes in life testing experiments. In the type-I censoring scheme, tested time is prefixed and the number of failure units is random. But, in type-II censoring scheme, the tested time is random and the number of failure units is prefixed. The two types of censoring have the lack of memory where the number of failure units may be very small or zero in type-I censoring scheme, but the total time of the test may be very large in type-II censoring scheme. And, the joint censoring scheme of type-I and type-II is called hybrid censoring scheme.In the plan of type-I hybrid censoring scheme (type-I HCS),n units are randomly selected from the product. The ideal test time and a suitable number of failure units that need statistical inference are proposed to be η∗ and m, respectively. The experimenter terminates the test at the min (η∗, Xm). Type-I HCS is exposed and studied by different authors, [1, 2] and recently by [3]. But, in the plan of type-II hybrid censoring scheme (type-II HCS), also, n identical units are randomly selected from the product. And, the ideal test time and a suitable number of failure units that need statistical inference are proposed to be η∗ and m, respectively. The experimenter terminates the test at the max (η∗, Xm); for more details, see [4]. The types of censoring, type-I HCS and type-II HCS also, satisfy the property that a smaller number of failures may be zero and there is a large test time, respectively; see [5]. Then, this problem has been treated with a generalized form of two types of censoring schemes known as a generalized hybrid censoring scheme (GHCS) [6].For type-I GHCS,n identical units selected from product to put to the test and two prefixed integers s and m satisfying that 1≤s<m≤n and prefixed time η∗∈0,∞ are proposed. The plan of type-I GHCS can be described as follows. When the experiment is running, the failure time is recorded until s number of failures is observed. If Xs<η∗, then the test is terminated at a minimum time of (η∗, Xm). In another case, if η∗<Xs<Xm, the test is terminated at Xs. Then, in type-I GHCS, the minimum number s of failure must be satisfied and the data is summarized as X=X1;n,X1;n,…,Xν;n where ν is defined by(1)ν=s,atη∗<Xs<Xm,m,atXs<Xm<η∗,ν,s<ν<matXs<η∗<Xm.For type-II GHCS,n identical units selected from the product to put to the test and prefixed integers m as well as two prefixed times η1∗,η2∗∈0,∞ satisfying η1∗<η2∗ are proposed. The plan of type-II GHCS is described as follows. After the experiment is running, the failure time is recorded until the time η1∗ is reached. If Xm<η1∗, then the test is terminated at η1∗. In another case, if η1∗<Xm<η2∗, the test is terminated at Xm. But, if η1∗<η2∗<Xm, then, the test is terminated at η2∗. Then, we say that the minimum time η1∗ must be observed and the maximum time η2∗ cannot be beyond it, and the random time data X=X1;n,X1;n,…,Xν;n where ν is defined by(2)ν=m,atη1∗<Xm<η2∗,ν,ν<matη1∗<η2∗<Xm,ν,ν>matXm<η1∗.In life testing experiments, the problem of obtaining sufficient information about the life of a product under recent technology is more difficult for a long-life product. Then, the related statistical inferences became more difficult. This problem can be solved with a good choice of the type of censoring scheme. Another solution to this problem is exposing the test unit to stress higher than normal stress conditions which is known as accelerated life tests (ALTs). Studies [7, 8] presented the key reference of ALTs. Recently, this problem is handled in [9, 10]. Different forms of ALTs are available. The first is known as constant stress ALTs, in which the experiment is loaded under constant stress until the final point of the experiments. The second type is called step stress ALTs, in which the experiment is running at different stress levels and changing at a prefixed time or number [11]. The last type is progressive stress ALTs, in which the stress is kept with a continuous increase at all experiment steps [12]. In some tests, units are running under normal stress and accelerated stress; then, this type of acceleration is called partially accelerated life test. The model of partially constant ALTs can be built as follows. For n identical tested units randomly chosen from population, n1 and (n−n1) units are selected randomly to test under used and accelerated conditions, respectively. Then, the failure times at each stage are recorded under a determined censoring scheme. Units under test can fail with different fetal risks; one of these risks is caused by the failure and the problem of measuring the risk of one cause of the failure known as competing risks. This problem is discussed by different authors [13–16]. Recently, this problem is handled for the accelerated model in [17].The main objective in this paper is adopting the type-I GHCS with a partially constant ALT model when test units fail with only two independent causes of failure and the failure time is distributed with Gompertz distribution (GD). The Gompertz lifetime population with random variableX has probability density function (PDF) given by(3)fx=θexpexpαx−θαexpαx−1,x>0,α,θ>0,and cumulative distribution function (CDF) is presented by(4)Fx=1−exp−θαexpαx−1,where θ and α are shape parameters. Then, we describe the mechanism of the model and formulate the likelihood function. Also, we present under observed data the point and interval estimators of model parameters with maximum likelihood and Bayes estimations. The theoretical results are measured and compared with Monte Carlo simulation and data analysis.The paper is organized as follows. Section2 presents some abbreviations and the model description. Section 3 gives the classical estimation with the MLE method. Section 4 presents the Bayes estimation with the MCMC method. Section 5 reported the results of the Monte Carlo studying. Section 6 presents lifetime data analysis for illustrating purpose. In Section 7, we give a report about the numerical results obtained from the simulation study and data analysis.


## 2. The Model

In this section, we present the list of abbreviations that are used in the paper as well as a complete description of the model mechanism and the corresponding likelihood function.


### 2.1. Abbreviations

GD : Gompertz distributionMLE : maximum likelihood estimationME : meanPC : probability coverageCDF : cumulative distribution functionHRF : hazard failure rate functionxij: failure time under stress i and cause jMH : Metropolis–Hastings algorithmCI : credible intervalsMCMC : Markov chain Monte CarloMSE : mean squared errorML : mean interval lengthPDF : probability density functionSF : survival functionSEL : squared error lossδj: the indicator value expressed to causeACI : approximate confidence intervalCDF : cumulative distribution function


### 2.2. The Model under Type-I GHCS

Suppose thatn identical units are randomly chosen from the population to put to the test, and let n1 be randomly selected to test under used condition and n2=n−n1 units to test under stress condition. Previously, two integers s and m such that s<m and m≤minn1,n2 with ideal test time η∗ are determined. Firstly, when the test is running, the times to failure Xij;nj, i=1,2,…,νj and j=1,2, denoting used and accelerated conditions, respectively, are recorded. The values of νj are defined in (1), and the mechanism is described as follows. If Xsj<η∗, then, the test is terminated at the minimum time of (η∗, Xmj). In another case, if η∗<Xsj<Xmj, the test is terminated at Xsj. Considering only the two causes of failure, the time to failure and the corresponding cause of failure are recorded. Then, type-I GHCS under the competing risks model is described by(5)X1j;nj,δ1j<X2j;nj,δ2j<⋯<Xνjj;nj,δνjj,where(6)νj=s,atη∗<Xsj<Xmj,m,atXsj<Xmj<η∗,ν,s<ν<matXsj<η∗<Xmj,and δij=1,0 means cause one or cause two, i=1,2,…,νj. The joint likelihood function of observed data {x1j;nj,δ1j<x2j;nj,δ2j<⋯<xνjj;nj,δνjj} with the CDF and PDF of random variables given byFljx and fljx, l=1,2, denotes cause 1 and cause 2, respectively; then, it is given by(7)Lφ¯|x¯=∏j=12Qj∏i=1νjf1jxij;njS2jxij;njδijf2jxij;njS1jxij;nj1−δij×S1jxνj;,njS2jxνj;njnj−νj,where Qj=nj!/nj−νj! and(8)δil=1,l=1,0,l=2.Considering that, tested units with CDF given by (4) for used condition and for independent two causes of failure that are reduced to the distribution have PDFs given by(9)f1lx=θlexpexpαx−θlαexpαx−1,l=1,2.The Gompertz lifetime distribution with common shape parametersα and different shape parameters θl, l=1,2 and also the CDFs and SFs are given by(10)F1lx=1−exp−θlαexpαx−1,(11)S1Lx=exp−θlαexpαx−1.Consider that the proportional hazard model (also named Cox model) is relevant to handle the effects of the environment or stress on the lifetime distribution. Thus, the survival functionS1l⋅ under used stress is GD but the survival function under the higher stress takes the form(12)S2lx=S1lxβ.Therefore,S2lx, CDFs, and PDFs under accelerated condition are given, respectively, by(13)S2lx=exp−βθlαexpαx−1,(14)F2lx=1−exp−βθlαexpαx−1,(15)f2lx=θlβexpexpαx−βθlαexpαx−1.


## 2.1. Abbreviations

GD : Gompertz distributionMLE : maximum likelihood estimationME : meanPC : probability coverageCDF : cumulative distribution functionHRF : hazard failure rate functionxij: failure time under stress i and cause jMH : Metropolis–Hastings algorithmCI : credible intervalsMCMC : Markov chain Monte CarloMSE : mean squared errorML : mean interval lengthPDF : probability density functionSF : survival functionSEL : squared error lossδj: the indicator value expressed to causeACI : approximate confidence intervalCDF : cumulative distribution function


## 2.2. The Model under Type-I GHCS

Suppose thatn identical units are randomly chosen from the population to put to the test, and let n1 be randomly selected to test under used condition and n2=n−n1 units to test under stress condition. Previously, two integers s and m such that s<m and m≤minn1,n2 with ideal test time η∗ are determined. Firstly, when the test is running, the times to failure Xij;nj, i=1,2,…,νj and j=1,2, denoting used and accelerated conditions, respectively, are recorded. The values of νj are defined in (1), and the mechanism is described as follows. If Xsj<η∗, then, the test is terminated at the minimum time of (η∗, Xmj). In another case, if η∗<Xsj<Xmj, the test is terminated at Xsj. Considering only the two causes of failure, the time to failure and the corresponding cause of failure are recorded. Then, type-I GHCS under the competing risks model is described by(5)X1j;nj,δ1j<X2j;nj,δ2j<⋯<Xνjj;nj,δνjj,where(6)νj=s,atη∗<Xsj<Xmj,m,atXsj<Xmj<η∗,ν,s<ν<matXsj<η∗<Xmj,and δij=1,0 means cause one or cause two, i=1,2,…,νj. The joint likelihood function of observed data {x1j;nj,δ1j<x2j;nj,δ2j<⋯<xνjj;nj,δνjj} with the CDF and PDF of random variables given byFljx and fljx, l=1,2, denotes cause 1 and cause 2, respectively; then, it is given by(7)Lφ¯|x¯=∏j=12Qj∏i=1νjf1jxij;njS2jxij;njδijf2jxij;njS1jxij;nj1−δij×S1jxνj;,njS2jxνj;njnj−νj,where Qj=nj!/nj−νj! and(8)δil=1,l=1,0,l=2.Considering that, tested units with CDF given by (4) for used condition and for independent two causes of failure that are reduced to the distribution have PDFs given by(9)f1lx=θlexpexpαx−θlαexpαx−1,l=1,2.The Gompertz lifetime distribution with common shape parametersα and different shape parameters θl, l=1,2 and also the CDFs and SFs are given by(10)F1lx=1−exp−θlαexpαx−1,(11)S1Lx=exp−θlαexpαx−1.Consider that the proportional hazard model (also named Cox model) is relevant to handle the effects of the environment or stress on the lifetime distribution. Thus, the survival functionS1l⋅ under used stress is GD but the survival function under the higher stress takes the form(12)S2lx=S1lxβ.Therefore,S2lx, CDFs, and PDFs under accelerated condition are given, respectively, by(13)S2lx=exp−βθlαexpαx−1,(14)F2lx=1−exp−βθlαexpαx−1,(15)f2lx=θlβexpexpαx−βθlαexpαx−1.


## 3. Estimation under ML Method

The results of the point and asymptotic confidence intervals of model parameters are discussed in this section with MLE for two independent causes of failure.Letx=X1j;nj,δ1j,X2j;nj,δ2j,…,Xνjj;nj,δνjj be the sample of type-I GHC competing risks data from GD; for distribution (10) and (14), the joint function (7) is reduced to(16)Lα,θ1,θ2,β|x¯=θ1m1θ2m2βν2expα∑i=1ν1xi1+∑i=1ν2xi2−θ1+θ2α×∑i=1ν1δi1expαxi1−1−βθ1+θ2α∑i=1ν2δi2expαxi2−1−θ1+θ2αexpαxν1−1−βθ1+θ2αexpαxν2−1,where ν1 and ν2 denote the number of unit failures under used and accelerated conditions, respectively. Also, m1=∑j=12∑i=1νjδij and m2=∑j=12∑i=1νj1−δij are the number of unit failures under the first and second causes, respectively. Then, the natural logarithm of the likelihood function (16) is reduced to(17)ℓα,θ1,θ2,β|x¯=m1logθ1+m2logθ2+ν2logβ+α∑i=1ν1xi1+∑i=1ν2xi2−θ1+θ2α∑i=1ν1δi1expαxi1−1−βθ1+θ2α×∑i=1ν2δi2expαxi2−1−θ1+θ2αexpαxν1−1−βθ1+θ2αexpαxν2−1.


### 3.1. Point Estimators

The point MLE of model parameters can be obtained after taking the partial derivatives of (17) with respect to vector φ=α,θ1,θ2,β. Then, the derivative with respect to θ1 and θ2 is reduced to(18)θ1α,β=αm1D,(19)θ2α,β=αm2D,(20)βα=ν2∑i=1ν1δi1expαxi1−1+expαxν1−1m1+m2−ν2∑i=1ν2δi2expαxi2−1+expαxν2−1,(21)∑i=1ν1xi1+∑i=1ν2xi2+θ1+θ2α2∑i=1ν1δi11−αxi1expαxi1−1+βθ1+θ2α2×∑i=1ν2δi21−αxi2expαxi2−1+θ1+θ2α21−αxν1expαxν1−1+βθ1+θ2α21−αxν2expαxν2−1=0,where(22)D=∑i=1ν1δi1expαxi1−1+β∑i=1ν2δi2expαxi2−1+expαxν1−1+βexpαxν2−1.Then, the likelihood equations are reduced to one linear equation ofα obtained after replacing θ1, θ2, and β by (18)–(20) in (21). The initial value of any iteration can be obtained from the profile likelihood function obtained from (16) after replacing θ1, θ2, and β. Then, the estimates are obtained α^,θ^1,θ^2, and β^.


### 3.2. Interval Estimation

The Fisher information matrix is defined as the minus expectation of second derivatives from the log-likelihood function with respect to model parameters. In practice, the expectation problem of the second derivative is more difficult to practice; then, the approximate Fisher information presents a suitable approximation that is used to build interval estimation as follows. LetΦ present the second derivative of parameters vector φ=α,θ1,θ2,β given by(23)Φ=−∂2ℓα,θ1,θ2,β|x¯∂φi∂φj.Then, the approximate information matrixΦ at the values of the MLE of the parameters vector φ is denoted by Φ0α^,θ^1,θ^2,β^ where φ=α,θ1,θ2,β. Then, the estimate φ^=α^,θ^1,θ^2,β^ is distributed as a normal distribution with mean φ=α,θ1,θ2,β and variance-covariance matrix Φ0α^,θ^1,θ^2,β^ described by(24)φ^∝Nα,θ1,θ2,β,Φ0−1α^,θ^1,θ^2,β^.Then,1001−2γ% approximate interval estimate of φ=α,θ1,θ2,β is given by(25)α^∓zγσ11,θ^1∓zγσ22,θ^2∓zγσ33,β^∓zγσ44,where zγ presents the standard normal values with probability tailed γ and the values σ11, σ22, σ33, and σ44 are the diagonal of the matrix Φ0−1α^,θ^1,θ^2,β^.


## 3.1. Point Estimators

The point MLE of model parameters can be obtained after taking the partial derivatives of (17) with respect to vector φ=α,θ1,θ2,β. Then, the derivative with respect to θ1 and θ2 is reduced to(18)θ1α,β=αm1D,(19)θ2α,β=αm2D,(20)βα=ν2∑i=1ν1δi1expαxi1−1+expαxν1−1m1+m2−ν2∑i=1ν2δi2expαxi2−1+expαxν2−1,(21)∑i=1ν1xi1+∑i=1ν2xi2+θ1+θ2α2∑i=1ν1δi11−αxi1expαxi1−1+βθ1+θ2α2×∑i=1ν2δi21−αxi2expαxi2−1+θ1+θ2α21−αxν1expαxν1−1+βθ1+θ2α21−αxν2expαxν2−1=0,where(22)D=∑i=1ν1δi1expαxi1−1+β∑i=1ν2δi2expαxi2−1+expαxν1−1+βexpαxν2−1.Then, the likelihood equations are reduced to one linear equation ofα obtained after replacing θ1, θ2, and β by (18)–(20) in (21). The initial value of any iteration can be obtained from the profile likelihood function obtained from (16) after replacing θ1, θ2, and β. Then, the estimates are obtained α^,θ^1,θ^2, and β^.


## 3.2. Interval Estimation

The Fisher information matrix is defined as the minus expectation of second derivatives from the log-likelihood function with respect to model parameters. In practice, the expectation problem of the second derivative is more difficult to practice; then, the approximate Fisher information presents a suitable approximation that is used to build interval estimation as follows. LetΦ present the second derivative of parameters vector φ=α,θ1,θ2,β given by(23)Φ=−∂2ℓα,θ1,θ2,β|x¯∂φi∂φj.Then, the approximate information matrixΦ at the values of the MLE of the parameters vector φ is denoted by Φ0α^,θ^1,θ^2,β^ where φ=α,θ1,θ2,β. Then, the estimate φ^=α^,θ^1,θ^2,β^ is distributed as a normal distribution with mean φ=α,θ1,θ2,β and variance-covariance matrix Φ0α^,θ^1,θ^2,β^ described by(24)φ^∝Nα,θ1,θ2,β,Φ0−1α^,θ^1,θ^2,β^.Then,1001−2γ% approximate interval estimate of φ=α,θ1,θ2,β is given by(25)α^∓zγσ11,θ^1∓zγσ22,θ^2∓zγσ33,β^∓zγσ44,where zγ presents the standard normal values with probability tailed γ and the values σ11, σ22, σ33, and σ44 are the diagonal of the matrix Φ0−1α^,θ^1,θ^2,β^.


## 4. Bayes Estimation

In this section, we present the Bayes point and interval estimation for the unknown model parameters to formulate the available information in the form of statistical distribution. The available information is exposed in the prior information and information exposed in the data. So, we consider the independent gamma priors for Gompertz parameters {α,θ1,θ2} and noninformative prior for accelerated factor β as follows:(26)α∝αa1−1exp−b1α,θ1∝θ1a2−1exp−b2θ1,θ2∝θ2a3−1exp−b3θ2,β∝β−1.Then, the joint prior density is given by(27)h∗α,θ1,θ2,β∝αa1−1θ1a2−1θ2a3−1β−1exp−b1α−b2θ1−b3θ2.The joint posterior distribution can be formulated by(28)hα,θ1,θ2,β|x¯=h∗α,θ1,θ2,βLα,θ1,θ2,β|x¯∫⋯∫h∗α,θ1,θ2Lα,θ1,θ2,β|x¯dαdθ1dθ2dβ.The Bayes estimate of model parameters depends on the posterior distribution and the choice of the loss function. Without loss of generality, considering squared error loss function (SEL), the Bayes estimators for any functiong (α,θ1,θ2,β) are formulated by(29)g^B=∫⋯∫gα,θ1,θ2,βhα,θ1,θ2,βdαdθ1dθ2dβ.Integration in (28) and (29) is generally more difficult; hence, we need some approximation to compute these integrations. Different methods are available to approximate the integral such as numerical integration from the important ones applied in the Bayes context called MCMC method described as follows.


### 4.1. MCMC Approach

The problem estimation with the Bayesian approach with the help of the MCMC method is needed to build the posterior conditional distributions of model parameters as follows:(30)hα,θ1,θ2,β|x¯∝αa1−1θ1m1+a2−1θ2m2+a3−1βν2−1exp−αb1−∑i=1ν1xi1−∑i=1ν2xi2−θ1+θ2α∑i=1ν1δi1expαxi1−1−βθ1+θ2α∑i=1ν2δi2×expαxi2−1−b2θ1−b3θ2−θ1+θ2αexpαxν1−1−βθ1+θ2αexpαxν2−1.Therefore, the problem of building conditional distributions of (28) given data is presented by(31)h1θ1|α,θ2,βx¯∝exp−θ1αb2α+∑i=1ν1δi1expαxi1−1+β∑i=1ν2δi2expαxi2−1+expαxν1−1+βexpαxν2−1θ1m1+a2−1,(32)h2θ2|α,θ2,βx¯∝exp−θ2αb3α+∑i=1ν1δi1expαxi1−1+β∑i=1ν2δi2expαxi2−1+expαxν1−1+βexpαxν2−1θ2m2+a3−1,(33)h3β|α,θ1,θ2,x¯∝βν2−1exp−βθ1+θ2α∑i=1ν2δi2expαxi2−1+expαxν2−1,(34)h4α|θ1,θ2,β,x¯∝αa1−1exp−αb1−∑i=1ν1xi1−∑i=1ν2xi2−θ1+θ2α×∑i=1ν1δi1expαxi1−1−βθ1+θ2α∑i=1ν2δi2×expαxi2−1−θ1+θ2αexpαxν1−1−βθ1+θ2αexpαxν2−1.The conditional distribution (31) to (34) shows that the conditional posterior distribution of θ1,θ2, and β takes gamma distribution. But, the conditional distribution of α is more similar to the normal distribution. Then, the suitable scheme of the MCMC method is Metropolis–Hastings (MH) under Gibbs algorithms [18] described as follows.


### 4.2. MCMC Algorithm

Step 1: begin with initial parameter valuesα0,θ10,θ20,β0=α^,θ^1,θ^2,β^ put κ=1.Step 2: gamma distribution is used to generateθ1κ from (29), θ2κ from (30), and βκ from (31) with the Gibbs technique.Step 3: normal proposal distribution is used to generateακ from the conditional distribution (32).Step 4: report the value of the parameters vectorφκ=ακ,θ1κ,θ2κ,βκ.Step 5: putκ to be κ+1.Step 6: repeat steps (2) to (5)N times.Step 7: determine the iteration number that is needed for the stationary stateN∗ (burn-in); then, for any function gα,θ1,θ2,β, Bayes estimators are presented by(35)g^B=1N−N∗∑i=N∗+1Ngi,and the corresponding variance is defined by(36)Vg=1N−N∗∑i=N∗+1Ngi−g^B2,wheregi=gαi,θ1i,θ2i,βi and i=N∗+1,2,…,N.Step 8: the ordered value of the vectorgi=gαi,θ1i,θ2i,βi is denoted by gi, and then, the corresponding 1001−2γ% credible interval of g is given by(37)gγN−N∗,g1−γN−N∗.


## 4.1. MCMC Approach

The problem estimation with the Bayesian approach with the help of the MCMC method is needed to build the posterior conditional distributions of model parameters as follows:(30)hα,θ1,θ2,β|x¯∝αa1−1θ1m1+a2−1θ2m2+a3−1βν2−1exp−αb1−∑i=1ν1xi1−∑i=1ν2xi2−θ1+θ2α∑i=1ν1δi1expαxi1−1−βθ1+θ2α∑i=1ν2δi2×expαxi2−1−b2θ1−b3θ2−θ1+θ2αexpαxν1−1−βθ1+θ2αexpαxν2−1.Therefore, the problem of building conditional distributions of (28) given data is presented by(31)h1θ1|α,θ2,βx¯∝exp−θ1αb2α+∑i=1ν1δi1expαxi1−1+β∑i=1ν2δi2expαxi2−1+expαxν1−1+βexpαxν2−1θ1m1+a2−1,(32)h2θ2|α,θ2,βx¯∝exp−θ2αb3α+∑i=1ν1δi1expαxi1−1+β∑i=1ν2δi2expαxi2−1+expαxν1−1+βexpαxν2−1θ2m2+a3−1,(33)h3β|α,θ1,θ2,x¯∝βν2−1exp−βθ1+θ2α∑i=1ν2δi2expαxi2−1+expαxν2−1,(34)h4α|θ1,θ2,β,x¯∝αa1−1exp−αb1−∑i=1ν1xi1−∑i=1ν2xi2−θ1+θ2α×∑i=1ν1δi1expαxi1−1−βθ1+θ2α∑i=1ν2δi2×expαxi2−1−θ1+θ2αexpαxν1−1−βθ1+θ2αexpαxν2−1.The conditional distribution (31) to (34) shows that the conditional posterior distribution of θ1,θ2, and β takes gamma distribution. But, the conditional distribution of α is more similar to the normal distribution. Then, the suitable scheme of the MCMC method is Metropolis–Hastings (MH) under Gibbs algorithms [18] described as follows.


## 4.2. MCMC Algorithm

Step 1: begin with initial parameter valuesα0,θ10,θ20,β0=α^,θ^1,θ^2,β^ put κ=1.Step 2: gamma distribution is used to generateθ1κ from (29), θ2κ from (30), and βκ from (31) with the Gibbs technique.Step 3: normal proposal distribution is used to generateακ from the conditional distribution (32).Step 4: report the value of the parameters vectorφκ=ακ,θ1κ,θ2κ,βκ.Step 5: putκ to be κ+1.Step 6: repeat steps (2) to (5)N times.Step 7: determine the iteration number that is needed for the stationary stateN∗ (burn-in); then, for any function gα,θ1,θ2,β, Bayes estimators are presented by(35)g^B=1N−N∗∑i=N∗+1Ngi,and the corresponding variance is defined by(36)Vg=1N−N∗∑i=N∗+1Ngi−g^B2,wheregi=gαi,θ1i,θ2i,βi and i=N∗+1,2,…,N.Step 8: the ordered value of the vectorgi=gαi,θ1i,θ2i,βi is denoted by gi, and then, the corresponding 1001−2γ% credible interval of g is given by(37)gγN−N∗,g1−γN−N∗.


## 5. Monte Carlo Simulation Study

In this section, we assess the developed results in classical MLE or Bayesian approach for different combinations of sample sizen and different combinations of n1 and n2. Also, the study reported different effect sizes s, m and different test times η∗. We adopt one parameter set to be α,θ1,θ2,β=0.1,0.05,0.08,2.0. For prior information, we adopt noninformative prior0 (posterior is proportional with likelihood function) and informative prior1 ai,bi=1,5,0.1,2,0.1,1. Mathematics Vr 10 is used, and iteration is reported for 1000 samples of type-I GHC data generated from the Gompertz distribution. For the Bayesian approach with MCMC methods, we generate N=11000 and discard the first N∗=1000. For the point estimate, we compute the mean value of parameter estimates (MEs) and mean squared error (MSE). But, interval estimation is measured with probability coverage and the mean interval length. The results of the simulation study are reported in Tables 1 and 2 as follows.Table 1
MEs and MSEs forα,θ1,θ2,β=0.1,0.05,0.08,2.0.
MLEMCMCprior0MCMCprior1η∗n,n1,n2,s,mPar.MEsMSEsMEsMSEsMEsMSEs5.0(70, 35, 35, 15, 30)α0.12310.08250.12020.08110.11870.0642θ10.06240.03420.06110.03250.06000.0227θ20.08720.05360.08510.05140.08170.0324β3.31241.22513.31001.22272.42501.0011(70, 35, 35, 25, 30)α0.11750.07240.11450.06920.11010.0592θ10.05840.03000.05710.02970.05420.0201θ20.08320.04820.08170.04910.08130.0287β2.81421.12432.78421.11452.2540.9852(70, 30, 40, 15, 30)α0.12500.08310.12150.08310.11990.0651θ10.06410.03570.06240.03420.06210.0240θ20.08930.05480.08670.05280.08250.0329β3.31391.22633.31151.22332.42621.0026(70, 30, 40, 25, 30)α0.11820.07310.11320.06990.11220.0601θ10.05910.03140.05820.02990.05560.0214θ20.08500.04800.08400.04810.08250.0298β2.81551.12512.78561.11522.25390.985710(70, 35, 35, 15, 30)α0.12410.07820.11990.07720.11010.0574θ10.06150.03020.05740.02890.05640.0200θ20.08600.04890.08410.04700.08210.0241β3.30020.99453.00010.98802.21350.8852(70, 35, 35, 25, 30)α0.11770.06230.11990.07110.10820.0452θ10.05620.02780.05410.02550.05220.0185θ20.08110.03500.08100.04420.08170.0211α2.71420.97992.64520.97772.21010.8800(70, 30, 40, 15, 30)α0.12710.07820.11990.07920.11170.0574θ10.06320.03210.05880.03010.05750.0215θ20.08710.04990.08570.04900.08290.0254β3.30220.99623.00250.98982.21280.8870(70, 30, 40, 25, 30)α0.11810.06360.12140.07180.10990.0460θ10.05770.02910.05620.02630.05380.0197θ20.08150.03700.08190.04600.08220.0215β2.71550.98122.64710.97722.21220.8817Table 2
MLs and PCs for 95% interval estimationα,θ1,θ2,β=0.1,0.05,0.08,2.0.
MLEMCMCprior0MCMCprior1η∗n,n1,n2,s,mPar.MLPCMLPCMLPC5.0(70, 35, 35, 15, 30)α0.4210.900.3990.900.2840.91θ10.1240.890.1340.910.1140.92θ20.2150.880.2000.900.1410.91β4.1250.904.1010.913.5210.96(70, 35, 35, 25, 30)α0.3210.900.3110.920.1840.92θ10.1010.910.0940.910.0890.92θ20.1870.900.1600.930.1180.92β3.4210.913.4050.913.0110.93(70, 30, 40, 15, 30)α0.4350.890.4120.900.2950.91θ10.1350.890.1550.890.1270.92θ20.2200.890.2140.910.1610.93β4.1310.894.1170.913.5280.90(70, 30, 40, 25, 30)α0.3410.900.3180.920.1970.93θ10.1180.900.1120.900.0960.91θ20.1990.900.1720.920.1240.92β3.4340.903.4140.913.0180.9110(70, 35, 35, 15, 30)α0.3140.900.3050.910.2000.93θ10.0950.910.0880.910.0780.92θ20.1470.900.1300.920.0950.91β4.0980.914.0010.913.3240.95(70, 35, 35, 25, 30)α0.2750.930.2660.920.1700.94θ10.0820.910.0760.930.0510.92θ20.0950.930.0810.920.0680.93β4.0720.913.8650.923.2140.94(70, 30, 40, 15, 30)α0.3310.910.3220.910.2110.91θ10.1120.900.0930.920.0850.94θ20.1620.900.1480.920.0990.94β4.1070.924.0140.903.3410.94(70, 30, 40, 25, 30)α0.2820.900.2710.900.1730.94θ10.0870.910.0850.930.0640.93θ20.1070.920.0920.910.0720.95β4.0780.923.8700.943.2230.92


## 6. Data Analysis Simulation

In this section, we choose the set of data generated from GD with respect to type-I GHCS and accelerated under partially constant stress ALTs. The random sample is generated from two GDs over the following algorithms.Step 1: let the total sample take the sizen=60 and n1=n2=30 and let s=15 and m=25.Step 2: suppose that the parameter vectors are randomly chosen to beφ=α,θ1,θ2,β=0.5,0.1,0.15,2.0; then, a suitable η∗ is chosen to be 2.5.Step 3: the prior information is almost selected to satisfyEφi≃ai/bi; then ai,bi=1,2,1,5,1,5,i=1,2,3.Step 4: generate two samples of sizen1=30 from the two populations (10) with causes of failure. The two samples are put in ordered pairs to choose the minimum from each pair. Then, the minimum values are put in ascending order and determine the sample with used conditions, random values, and its cause of failure. If Xs<η∗, then we terminated time min (η∗, Xm). In another case, if η∗<Xs<Xm, the terminated time is Xs. Then, the value ν1 is observed.Step 5: step 4 is repeated for distribution (13) to determine ν2.Step 6: computem1=∑j=12∑i=1νjδij and m2=∑j=12∑i=1νj1−δij.The data under used and accelerated conditions with its cause of failure is obtained from Tables3 and 4 with ν1=18 and ν2=25. The point MLE and corresponding confidence interval are summarized in Table 5. Also, for the Bayesian approach, we run the chain 11000 with the first 1000 values as burn-in, and the point and interval estimates are summarized in Table 5. Figures 1–8 describe the generated MCMC sample which describes the convergence satisfied by the MCMC method. The results obtained from each figure have shown that the MCMC method serves very well.Table 3
The generated data under used conditions.
0.18440.19000.20270.58290.70320.91031.27551.45151.57331.670000111000001.7471.75291.79581.99912.06562.36612.37732.4582.54262.996111000101003.35523.39243.41993.44123.46523.87784.19084.26064.8114.94161101101000Table 4
The generated data under accelerated conditions.
0.07460.12330.12980.15960.19500.38450.44210.50800.65620.807311111011000.90450.96501.12421.37391.41211.54161.58921.61911.64751.670501011100001.73881.79091.89322.18562.20122.28662.28702.63322.64702.79820001100001Table 5
Point and 95% confidence and credible intervals (ACIs and CIs) of MLE Bayes estimates.
Pa.s⋅ML⋅BMCMC95% ACIsLength95% CIsLengthα=0.50.74310.7271(0.2894, 1.1967)0.9073(0.3131, 1.1825)0.8694θ1=0.10.28080.2689(0.0454, 0.5162)0.4708(0.0644, 0.5542)0.3898θ2=0.150.32290.3076(0.0571, 0.5888)0.5317(0.1327, 0.5749)0.4422β=2.01.49761.7884(0.5732, 2.4220)1.8488(0.8346, 3.4357)2.6011Figure 1
Simulation number ofα generated by the MCMC method.Figure 2
The histogram ofα generated by the MCMC method.Figure 3
Simulation number ofθ1 generated by the MCMC method.Figure 4
The histogram ofθ1 generated by the MCMC method.Figure 5
Simulation number ofθ2 generated by the MCMC method.Figure 6
The histogram ofθ2 generated by the MCMC method.Figure 7
Simulation number ofβ generated by the MCMC method.Figure 8
The histogram ofβ generated by the MCMC method.


## 7. Conclusions

Modern technology products have a long period of time, and information about the life product is more difficult. Then, to overcome this problem, the experimenter determined the censoring scheme that serves this problem. In this paper, we choose type-I GHCS which keeps the minimum number needed in statistical inference in a small period of time. Also, this type of censoring is applied with the concept of partially constant ALTs for units that fail under two independent causes of failure and units that have Gompertz lifetime distribution. Moreover, we have seen that the proposed model can be easily extended for different populations. Also. we can mention that we can use a more general class of prior information such as priors with log-concave density functions. The simulation study is conducted, and the results are reported in Tables1 and 2 which show the following:(1)
The proposed model is more acceptable(2)
The results for the large value ofη∗ are more acceptable in terms of MSEs, PCs, and AL(3)
The results are getting better for increasing values of increasing (s, m)(4)
The results are getting better for closed values of (n1, n2)(5)
The results under MLEs and noninformative priors are closed(6)
Bayesian estimation under informative prior is better than MLEs(7)
The results for the selected set of parameters are more acceptable



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