# Inferences for Joint Hybrid Progressive Censored Exponential Lifetimes under Competing Risk Model

Authors: Abduallah M. Almarashi; Ali Algarni; A. M. Daghistani; G. A. Abd-Elmougod; S. Abdel-Khalek; Mohammad Z. Raqab
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The aim of this paper is devoted to the problem of comparative life tests under joint censoring samples from an exponential distribution with competing risks model. This problem is considered under the consideration that only two causes of failure are occurring and the units come from two production lines such that the exponential failure time of units is censored under a hybrid progressive Type-I censoring scheme. Maximum likelihood estimation and different Bayes methods of estimation are discussed. The asymptotic confidence intervals as well as the Bayes credible intervals are established. A real data set representing time to failure on two groups of strain male mice receiving radiation is analyzed for illustrative purposes. All theoretical results are assessed and compared through the Monte Carlo study.


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## Body

## 1. Introduction

Under the joint progressive censoring (JPC), the products from two different production lines, say Population-η1 (Pop-η1) and Population-η2(Pop-η2), are combined and put on a life testing experiment. Then, comparative life testing is done on these products to measure the relative merits of two competing durations of life products. Suppose that the units are produced from two different lines (Pop-η1 and Pop-η2) under the same conditions. Then, the experimenter can randomly select the two independent samples to put under time testing experiment. Due to the time and cost constraints, the termination of the statistical experiment arises naturally in various fields such as reliability, survival, and medical studies. For a detailed study of this type of data and related inferences, one may refer to Rao et al. [1], Basu [2], Johnson and Mehrotra [3], Mehrotra and Johnson [4], Bhattacharyya and Mehrotra [5], and Mehrotra and Bhattacharyya [6]. Under the JPC scheme, the two samples from Pop-η1 and Pop-η2 of sizes m and n, respectively, are combined and put on a life testing experiment. Balakrishnan and Rasouli [7] and Rasouli and Balakrishnan [8] introduced the JPC scheme when the data can be produced from two exponential populations. Shafay et al. [9] discussed the Bayesian inference based on a jointly Type-II censored sample from two exponential populations. Mondal and Kundu [10] addressed the problem of point and interval estimation of the unknown parameters of two Weibull distributions based on the Bayesian approach.In jointly Type-II censoring scheme, suppose that a sample of sizeD is taken from the first production line, Pop-η1, with cumulative distribution function (CDF) F1 and probability density function (PDF) f1, say, X1,X2,…,XD. Also, another independent sample of size N is taken from the second production line, Pop-η2, with CDF F2 and PDF f2, say, Y1,…,YN. Let m be the total number of observed failures in the experiment. Then, the ordered lifetime sample T1,T2,…,Tm forms a combined sample taken from both samples X1,X2,…,XDm and Y1,Y2,…,YNm with m=Dm+Nmbeing the joint Type-II censoring sample. Hence, the observed data consists of T,ω=T1,ω1,T2,ω2,…,Tm,ωm, 1≤m≤D+Nwith ωi=1 or 0 accordingly as Ti is taken from an X−or Y−sample, respectively. Let us denote the number of failed units from Pop-η1 and Pop-η2 by n1=∑i=1mωi and n2=m−n1=∑i=1m1−ωi, respectively. Then, the likelihood function based on a JPC sample is given by(1)LT,ω=D!N!D−n1!N−n2!∏i=1mf1tiωif2ti1−ωiS1tmD−n1S2tmN−n2,where S1. and S2. are the reliability functions of X and Y, respectively.In the conventional Type-I censoring scheme, the experiment continues up to the prespecified timeτ. On the other hand, the conventional Type-II censoring scheme requires the experiment to continue until a prespecified number of failures occur. If a mixture of Type-I and Type-II censoring schemes is applied, then the resulting censoring scheme is called a hybrid censoring scheme (HCS). In Type-I HCS, the test is terminated when a prefixed number, m<D+N, out of D+N items has failed, or when a prefixed time, τ, has been reached. In other words, the life test is terminated at a random time T∗=minTm,τ. In Type-II HCS, the experiment is stopped when T∗=maxTm,τ is reached.Moreover, the progressive censoring scheme can be adopted combined with a hybrid scheme to define Type-I hybrid progressive censoring scheme (Type-I HPCS), and it is described as follows. Suppose that a sample of sizen is randomly selected from the products to put under test and given an integer m, ideal test time τ, and censoring scheme R=R1,R2,…,Rm such that n=m+∑i=1mRi are determined. At each failure time Ti,Ri survival units are removed from the test, where i=1,2,…,J and 1≤J≤m. The experiment is terminated under Type-I HPCS at T∗,which is defined by T∗=minTm,τ. In this work, we focus our attention on Type-I HPCS. The HCS is quite useful in the reliability acceptance test. See, for example, Childs et al. [11] and Jeong et al. [12] for some development of the hybrid censored sampling plan for the exponential lifetime distributions.In real-life testing (medical and survival studies), it is quite common that more than one risk factor may be present at the same time, and only the smallest failure time can be observed when multiple modes of failure work together. An investigator is often interested in the assessment of specific risk in the presence of other risk factors. For this reason, the observed data consist of a failure time and an indicator denoting the cause of failure, and this model is known as the competing risks model. This aspect is discussed early by Cox [13] using the exponential data. Different properties of the competing risks model are exposed by Crowder [14], Balakrishnan [15], and Bakoban and Abd-Elmougod [16]. Recently, the properties of a competing risks model under the accelerated life test model are discussed by Ganguly and Kundu [17] and Algarni et al. [18]. In a random sample of size n of the life products and predetermined integer munder the competing risks model, the two causes of failure are independent. If δi denotes the cause of failure of the i-th ordered unit, then in this particular case, δi can take only two values δi=k,k=1,2. Suppose that, for given n and m, the failure time and the corresponding cause of failure are recorded; say Ti,δi,i=1,2,…,mis observed. For a given censoring scheme R=R1,R2,…,Rm, the final censored competing risk sample is abbreviated as(2)t,δ,R=t1,δ1,R1,t2,δ2,R2,…,tm,δm,Rm.Then, the likelihood function can be expressed as(3)Lt,δ,R=n!S1tmS2tmn−mn−m!∏i=1mf1tiS2tiρδi=1f2tiS1tiρδi=2S1tiS2tiRi,where(4)ρδi=k=1,δi=k,0,δi≠k,with k=1,2 and 0<t1<t2<⋯<tm<∞.The aim of this paper is to provide an analysis of Type-I HPC competing risks data. In latent failure time modeling, it is assumed that competing causes of failures are independent random variables. In this paper, it is assumed that the lifetimes of the competing causes of failure follow an exponential distribution with different scale parameters. The frequentist and Bayesian methods are applied for estimating the parameters involved in the underlying model. Because of the very complicated nature of the distribution functions, we propose the asymptotic confidence intervals (CIs) and two Bayes credible intervals (CrIs). The paper is summarized as follows. In Section2, the model assumptions and descriptions are presented. The maximum likelihood estimators (MLEs) and the corresponding CIs are developed in Section 3. Also, point and interval estimation using the Bayes approach is discussed in Section 4. In Section 5, we analyze a real data set for illustrative purposes and perform a numerical simulation study for assessing the so-developed methods. Finally, our findings and brief comments are summarized in Section 6.


## 2. The Model Description

Suppose that the joint sample of sizen=N+D of units is randomly selected from the two different production lines (Pop-η1, Pop-η2) under the same facility. The sample of size D is selected from the first production line (Pop-η1), and another independent sample of size N is selected from the second production line (Pop-η2). Suppose that integer m denotes the number of failures needed for statistical inferences, τ is the ideal test time, and the experiment is running under a censoring scheme R=R1,R2,…,Rm. When the first failure T1 is observed, its corresponding type ω1 (line-η1 or line-η2) and failure cause δ1 (first or second cause of failure) are observed; say T1,ω1,δ1 is recorded. At the second failure T2, we also observe ω2 and δ2, and then, T2,ω2,δ2 is recorded. The experiment continues until the minτ,Tm is observed. Under this censoring scheme, we have one of the following two types of observations:(5)Case I:t1<t2<⋯<tm,Case II:t1<t2<⋯<tJ,for1≤J<m,where J denotes the number of failures observed before time τ. The experiment is terminated at τ if τ<Tm and terminated at Tm if τ>Tm. Note that, in Case I, we may consider J=m, and for Case II, 1≤J<m. Therefore, the joint Type-I HPCS competing risks sample is(6)t,ω,δ=t1,ω1,δ1,t2,ω2,δ2,…,tJ,ωJ,δJ,1≤J≤m.Let the number of units failed from Pop-η1 be denoted by m1=∑i=1Jωi, and let the number of units failed from Pop-η2 be m2=∑i=1J1−ωi. For the two causes of failure, j=1,2, we assume that n1j=∑i=1mωi∗ρδi=j, the number of failed units coming from the Pop-η1 with the cause j. Also, n2j=∑i=1m1−ωj∗ρδi=j, the number of failed units coming from Pop-η2 with the cause j,j=1,2. Based on the exponential lifetimes’ data, we report the following assumptions:(1)
The failure timesTi,1,2,…,J are defined as Ti=minTik1,Tik2 with Tikj being the i−th failure time from the k−th population k=1,2 under cause jj=1,2.(2)
With parametersθkj, the random variable Tikjk=1,2andj=1,2 has the exponential distribution with PDF and CDF given, respectively, by(7)fkjt=θkjexp−θkjt,t>0,θkj>0,k=1,2andj=1,2,(8)Fkjt=1−exp−θkjt.The corresponding reliability and hazard rate functions are, respectively, given by(9)Skjt=exp−θkjt,hkjt=θkj,wherek and j denote the type and cause of the failure, respectively.(3)
The latent failure time is distributed with exponential distribution withθk1+θk2 scale parameters.(4)
The discrete random variablesnk1 and nk2 which describe the number of failed units under the first and second causes of failure are distributed as binomial distributions with sample size J−mk and probability of success θk1/θk1+θk2 and θk2/θk1+θk2, respectively.Based on these assumptions, the joint likelihood function of the observed samplet,ω,δ is given by(10)Lt,ω,δ|ϕ∝∏i=1Jh11tiρδi=1h12tiρδi=2S11tiS12tiωiS11tiS12tiR1i×h21tiρδi=1h22tiρδi=2S21tiS22ti1−ωiS21tiS22tiR2i×S11t∗S12t∗D−m1S21t∗S22t∗N−m2,where R1i+R2i=Ri with R1i and R2i being the number of censored items from the two production lines, Pop-η1 and Pop-η2, respectively, φ=θ11,θ12,θ21,θ22,(11)t∗=τ,ifτ<Tm,Tm,ifτ>Tm,0<t1<t2<⋯<tJ<∞.


## 3. Maximum Likelihood Estimation

Here, in this section, we obtain the MLEs of all parameters involved in the previous model based on the times to failure under Type-I HPC with two factors including the failure type and failure cause. Further, the asymptotic CIs for the model parameters are also obtained. The likelihood function based on the exponential failure times with PDF and CDF given in (7) and (8) can be expressed in the following form:(12)Lθ11,θ12,θ21,θ22|ϕ∝θ11n11θ12n12θ21n21θ22n22exp−θ11+θ12∑i=1JR1i+ωiti−D−m1θ11+θ12t∗−N−m2θ21+θ22t∗−θ21+θ22∑i=1JR2i+1−ωiti.From (12), we immediately obtain the log-likelihood function as(13)ℓθ11,θ12,θ21,θ22|t¯∝n11logθ11+n12logθ12+n21logθ21+n22logθ22−θ11+θ12×∑i=1JR1i+ωiti−θ21+θ22∑i=1JR2i+1−ωiti−D−m1θ11+θ12+N−m2θ21+θ22t∗.The partial derivatives of the log-likelihood function (13) are reduced to(14)∂ℓθ11,θ12,θ21,θ22|t¯∂θkj=0,k,j=1,2,which is readily reduced to(15)θ^1j=n1j∑i=1JR1i+ωiti+D−m1t∗,j=1,2,(16)θ^2j=n2j∑i=1JR2i+1−ωiti+N−m2t∗,j=1,2.Remark 1.
Under the settingn1j=∑i=1Jωi∗ρδi=j=0 or Dand n2j=∑i=1m1−ωi∗ρδi=j=0 or N, equations (15) and (16) do not lead to the existence of the MLEs of θ1j,θ2j,j=1,2. Further, the exact probability distributions of the estimators θ^1j and θ^2j are defined as the mixture of discrete and continuous distributions, which are difficult to be derived (see, for example, [19]).
From the joint log-likelihood function given by (12), the second partial derivatives are derived to be(17)∂2ℓθ11,θ12,θ21,θ22|t¯∂θ1j2=−n1jθ1j2,j=1,2,(18)∂2ℓθ11,θ12,θ21,θ22|t¯∂θ2j2=−n2jθ2j2,j=1,2,(19)∂2ℓθ11,θ12,θ21,θ22|t¯∂θij∂θkl=0,For eachij≠kl.
Taking the minus expectation of equations (18)–(20), we readily obtain the Fisher information matrix of the parameters θ11,θ12,θ21, and θ22, say λθ11,θ12,θ21,θ22, where(20)λθ11,θ12,θ21,θ22=n11θ1120000n12θ1220000n21θ2120000n22θ222at⟶θ^11,θ^12,θ^21,θ^22.
Then, the inverse Fisher information matrix under non-zero values of the elements of the diagonal elements provides the variance-covariance matrix of the MLEs. Hence, the approximate (1−2α)% CIs of the parameters θ11,θ12,θ21, and θ22, using the fact that θ11,θ12,θ21,θ22 is approximately normal distribution with mean θ11,θ12,θ21,θ22 and covariance matrix λ−1θ11,θ12,θ21,θ22, are established by(21)θ^11∓zαθ^11n11,θ^12∓zαθ^12n12,θ^21∓zαθ^21n21,θ^11∓zαθ^22n22,where zα is the upper 100α−th percentile of standard normal distribution.


## 4. Bayesian Estimation

Here, we discuss the Bayesian analysis of the model parameters based on the Type-I HPC competing risk sample. Both point estimation and credible interval estimation of the unknown parameter vectorφ=θ11,θ12,θ21,θ22are considered. Now, we specify the prior distributions of the parameters. It is desirable that the model parameters are independent such that all prior and posterior densities belong to similar families. These prior choices allow the posterior distribution to be analytically tractable and computationally efficient. A natural choice for the priors of φ1,φ2,φ3,φ4=θ11,θ12,θ21,θ22 would be to assume that the four quantities are independent gamma Gai,bi,i=1,2,3,4 distributions, respectively, with the following densities:(22)πiφi=biaiΓaiφiai−1exp−biφi,φ>0,where φi, and ai,bi are chosen to reflect prior knowledge about φi,i=1,2,3,4. The joint prior density of (22) is represented as(23)πϕ=∏i=14biaiΓaiϕiai−1exp−biϕi.By combining the joint prior density (23) and likelihood function (12), we have the joint posterior density πφ|t,ω,δ as follows:(24)πϕ|t,ω,δ∝b1a1b2a2b3a3b4a4Γa1Γa2Γa3Γa4θ11n11+a1−1θ12n12+a2−θ21n21+a3−1θ22n22+a4−1exp−b1θ11−b2θ12−b3θ21−b4θ22−θ11+θ12∑i=1JR1i+ωiti−θ21+θ22∑i=1JR2i+1−ωiti−D−m1θ11+θ12+N−m2θ21+θ22t∗.Hence, the joint posterior density can be expressed as the product of independent gamma densities as follows:(25)πϕ|t,ω,δ∝∏i=14ϕiAi−1exp−Biϕi,where(26)A1=n11+a1,A2=n11+a2,A3=n21+a3,A4=n22+a4,with(27)B1=b1+∑i=1JRi+ωiti+D−m1t∗,B2=b2+∑i=1JRi+ωiti+D−m1t∗,B3=b3+∑i=1JRi+1−ω1ti+N−m2t∗,B4=b4+∑i=1JRi+1−ω1ti+N−m2t∗.This turns out that the Bayes estimators can be determined based on a selected loss function. It is well known that the error and loss functions play an important role in Bayesian estimation. One of the most popular symmetric loss functions is the squared error (SE) loss function which gives equal weight to overestimation as well as underestimation. In some cases, the use of symmetric loss function may be inappropriate; see, for example, Basu and Ebrahimi [20]. For this reason, asymmetric loss functions can be proposed. One of the well-known asymmetric loss functions is the linear exponential (LINEX) loss function which was introduced by Varian [21]. Another alternative function is the general entropy (GE) loss function (see Ren et al. [22]). For estimating any parameter (say, φi) by a decision rule φ^i, using the Bayes approach, the following loss functions are considered.(1)
Squared error (SE) loss function is defined by the following function:(28)g1φi,φ^i=φ^i−φi2.Then, the Bayes estimatorsφ^iB of φi under SE loss function are given by(29)φ^iB=Eπφi|t,ω,δ.(2)
Linear exponential (LINEX) loss function is defined by(30)g2ϕi,ϕ^i=expcϕ^i−ϕi−cϕ^i−ϕi−1,c≠0.Then, the Bayes estimatorsφ^iB of φi under LINEX loss function are obtained to be(31)φ^iB=−1clogEπexp−cφi|t,ω,δ.(3)
General entropy (GE) loss functions are defined by(32)g3φi,φ^i=φ^iφiq−qlogφ^iφi−1,q≠0.Then, the Bayes estimatorφ^iB of φi under GE loss is(33)φ^iB=Eπωi−q|t,ω,δ−1/q.In our setting, we can express the Bayes estimators (BEs) under the SE, LINEX, and GE loss functions as follows.(1)
Under SE loss function, the BEs of model parameters are given by(34)φ^iB=AiBi.(2)
Under the LINEX loss function, the BEs of the model parameters are given by(35)ϕ^iB=−AiclogBiBi+c.(3)
Under the GE loss function, the BEs of the model parameters are given by(36)φ^iB=1BiΓAi−qΓAi−1/q.Clearly, the joint posterior density in (24) can be written as a product of four independent gamma density functions. Hence, the CrIs of parameters vector φ=θ11,θ12,θ21,θ22 depend on the distribution of the pivotal Ψi=2φ^iBi,i=1,2,3,4. The values Ψi are greater than zero and have χ2 distribution with 2Ai degrees of freedom (see Kundu and Joarder [19]). Then, 1−2α% CrIs of the parameters vector (φ) are given by(37)χ2Ai,α22Bi,χ2Ai,1−α22Bi,i=1,2,3,4.


## 5. Data Analysis and Simulation Study

Now, we perform an analysis of a real data set for illustrative purposes and compare the performances of the different methods of estimation based on Monte Carlo simulations.


### 5.1. Data Analysis

Let us consider the real data sets given in Hoel [23]. The data given in Table 1 were obtained from a laboratory experiment on two groups of strain male mice which had received a radiation dose of 300r at an age of 5-6 weeks. The first group of mice (Group-1) lived in a conventional laboratory environment while the second group (Group-2) was in a germ-free environment. Two major causes of death are considered: thymic lymphoma and reticulum cell sarcoma for Cause 1 and Cause 2, respectively. Therefore, we consider that failures of the first group come from the Pop-η1 and the failures of the second group come from the Pop-η2. For more convenience, let us divide the data points by 1000, which will not affect the inference procedures. Under consideration that m=60, τ=0.380, and R=0,0,…,0,68, the jointly Type-I HPCS competing risks random sample is presented in Table 2. The data reported in Table 2 show that J=56,m1,m2=33,23 and n11,n12,n21,n22=17,16,19,4. The random combined sample is selected from both populations, Pop-η1 and Pop-η2, of sizes D=61 and N=67, respectively. For Bayesian analysis, since we do not have any prior information about the model parameters, we consider the noninformative prior by setting ai=bi=0.0001,i=1,2,3,4. Hence, the MLEs and BEs under SE, LINEX, and GE losses and the corresponding CIs are computed and presented in Table 3. It can be observed from Table 3 that all point estimates are lying within the 95% asymptotic CIs and Bayes CrIs.Table 1
Time to failure of male mice which received a radiation dose at age 5-6 weeks.
Group 1Thymic lymphoma159189191198200207220235245250256261265266280343356383403414428432Other cases40425162163179206222228249252282324333341366385407420431441461462482517517524564567586619620621622647651686761763Group 2Thymic lymphoma158192193194195202212215229230237240244247259300301321337415434444485496529537624707800Other cases13624625537642156561661765265565866066267568173473673775776977780080782585585786486887087388289591093494210151019Table 2
The Type-II HPC competing risks sample for radiation dose data withm=60.
ωi11112211111222ti0.0400.0420.0510.0620.1360.1580.1590.1630.1790.1890.1910.1920.1930.194δi00000110011111ωi21121122111221ti0.1950.1980.2000.2020.2060.2070.2120.2150.2200.2220.2280.2290.2300.235δi11110111100111ωi22212211121211ti0.2370.2400.2440.2450.2460.2470.2490.2500.2520.2550.2560.2590.2610.265δi11110101001111ωi11122211211112ti0.2660.2800.2820.3000.3010.3210.3240.3330.3370.3410.3430.3560.3660.376δi11011100101100Table 3
The MLEs and BEs of the parameters and corresponding CIs.
MLSELLINXGE95% asymptotic CI95% Bayes CrIsc = 1.0c = −1.0q = 2.0q = −2.0θ110.93920.93920.91420.96610.85590.9664(0.4927, 1.3856)(0.5984, 1.3425)θ120.88390.88390.86040.90930.80060.9111(0.4508, 1.3170)(0.5544, 1.2760)θ210.85440.85450.83570.87420.78660.8766(0.4702, 1.2385)(0.5595, 1.2002)θ220.17990.17990.17590.18400.11020.2011(0.0036, 0.3561)(0.0614, 0.3487)


### 5.2. Simulation Study

Now, we compare the performances of the different methods of estimation based on Monte Carlo simulations. We compare the performances of the MLEs and BEs in terms of biases and mean square errors (MSEs). In this simulation, two sets of parameter values,φ=1.0,0.8,1.5,1.2 and 0.5,0.3,0.6,0.2, are considered. Here, we consider different effective sample sizes m=30,40,60 and different censoring schemes. In this study, we check how the effect of changing the sample size D+N,m censoring scheme R and the ideal test time τ is. Under the noninformative and informative priors, the biases and MSEs of the estimates over 1000 replications are computed for various censoring schemes. The algorithm used for the numerical computations can be described as follows:(1)
For given prior informationai,bii=1,2,3,4, the true value of the parameter vector φ=θ11,θ12,θ21,θ22 is selected almost to satisfy φi=Eφi=ai/bi.(2)
For givenR, the value R1i is generated from Ri binomial variant with the probability of success being equal to 1/2 and R2i=Ri−R1i.(3)
For givenm, generate m1 from m binomial distribution with probability of success being equal to 1/2. Then, m2=m−m1.(4)
Generate Type-I HPCS competing risks samples of sizemk from exponential distributions with parameter θk1+θk2,k=1,2.(5)
For givenτ and the joint Type-I HPC competing risks sample, determine the joint hybrid progressive Type-I competing risks sample.(6)
From the joint Type-I progressive competing risks, censoring sample determinesm1 and m2.(7)
The discrete random variablesnkj,k,j=1,2, are generated from binomial distributions with sample sizes m1 and m2 with the probability of success θk1/θk1+θk2 and θk2/θk1+θk2, respectively.(8)
The point and interval estimates based on the likelihood and Bayes approaches are computed.(9)
For comparison purposes, the average estimates (AEs) and MSEs are computed. Furthermore, the average lengths (ALs) and coverage probabilities (CPs) are evaluated.(10)
The simulation study is conducted using censoring schemes (CSs) described as follows:(38)CS=I:Ri=0,0,…,0,D+N−m,II:Ri=D+N−m,0,0,…,0,III:Ri=0,i≠m+12andmodd ori≠m2andmeven,D+N−m,i=m+12andmodd ori=m2andmeven,IV:Ri=0,i>D+N−m2,2,i≤D+N−m2.(11)
All results of the simulation study are reported in Tables4–7 for noninformative prior P0=ai,bii=1,2,3,4=0.0001,0.0001 and informative prior P1.Table 4
The average estimates and MSEs whenφ=1.0,0.8,1.5,1.2.
(D, N,m, τ)SCMLEBayes (P0)Bayes (P1)θ11θ12θ21θ22θ11θ12θ21θ22θ11θ12θ21θ22(30, 30, 30, 1.2)I1.2311.1241.6211.4251.2111.1121.6071.4091.1870.9821.58721.3140.2410.2040.4210.3740.2170.1920.4030.3550.1890.1140.3540.301II1.1991.1111.6041.4131.1871.0951.5891.3391.1420.9411.58391.2870.1920.1520.3720.3270.1780.1510.3490.3040.1410.0970.3040.251III1.2161.1071.6141.4131.2021.1031.5931.4011.1680.9711.5861.3030.2240.1900.4040.3550.2000.1750.3840.3370.1710.1010.3360.289VI1.2071.1111.6171.4081.2051.1091.5891.4041.1540.9591.5811.2980.2130.1970.4010.3420.1920.1640.3790.3310.1660.1090.3280.281(30, 30, 40, 1.2)I1.1321.0451.5871.3561.1221.0221.5141.3011.0890.9021.5271.2290.1670.1180.3480.3010.1480.1230.3180.2880.1150.0690.2890.214II1.1191.0101.5651.3391.1041.0121.5101.2891.0660.8891.5011.2070.1140.0850.3010.2510.1000.0800.2790.2420.0890.0540.2390.179III1.1191.0391.5811.3441.1181.0171.5071.3031.0810.8971.5191.2210.1490.1040.3310.2910.1320.1070.3040.2790.1000.0610.2790.211VI1.1211.0321.5781.3391.1211.0241.5131.3081.0780.8821.5081.2150.1380.1010.3240.2840.1190.1040.3080.2660.0940.0630.2810.198(50, 50, 60, 1.2)I1.1040.9821.5421.3141.1100.9781.5291.3031.0610.8751.5111.2090.0890.0610.2420.2040.0870.0740.2520.2140.0830.0540.2100.129II1.0990.9791.5281.3141.1070.9691.5111.3121.0580.8711.5031.2040.0720.0540.2090.1870.0820.0690.2250.1820.0710.0520.2070.114III1.1110.9791.5331.3101.1120.9711.5241.3111.0580.8711.5141.2120.0840.0580.2150.2060.0810.0710.2480.2090.0810.0550.2100.114VI1.1130.9721.5301.3071.1150.9821.5211.3081.0530.8661.5021.2040.0810.0540.2120.2080.0770.0750.2420.2030.0750.0510.2070.105(30, 30, 40, 2.2)I1.1121.0241.5651.3331.1011.0021.5011.2891.0700.8821.5111.2120.1450.1010.3220.2820.1250.1020.3000.2650.1010.0480.2650.191II1.1111.0011.5571.3251.0821.0071.5031.2781.0580.8721.4901.1890.0920.0960.2850.2360.0890.0760.2630.2310.0800.0510.2280.165III1.1201.0291.5751.3401.1071.0111.4971.2881.0640.8831.5011.2030.1240.0810.3040.2690.10110.0890.2810.2520.0830.0470.2640.190VI1.1051.0211.5491.3271.1091.0011.4951.2891.0560.8541.2891.2030.1190.0910.3010.2640.0870.0940.3000.2490.0880.0590.2620.182(50, 50, 60, 2.2)I1.0890.9751.5291.3011.1030.9641.5141.2851.0450.8651.5031.2010.0890.0610.2420.2040.0870.0740.2520.2140.0830.0540.2100.129II1.0810.9681.5191.3031.1020.9481.5011.3041.0510.8591.5101.2010.0680.0490.1880.1550.0710.0610.2110.1690.0640.0490.2010.101III1.1030.9681.5251.3041.1030.9641.5131.3021.0510.8681.5011.2030.0750.0510.2000.1880.0720.0610.2190.1870.0730.0520.2020.103VI1.1090.9651.5241.3021.1040.9771.5141.3031.0440.8581.4921.2030.0740.0470.2000.1890.0620.0640.2240.1810.0630.0490.2010.099Table 5
The ALs and CPs of95% CIs when φ=1.0,0.8,1.5,1.2.
(D, N,m, τ)SCMLEBayes (P0)Bayes (P1)θ11θ12θ21θ22θ11θ12θ21θ22θ11θ12θ21θ22(30, 30, 30, 1.2)I2.3151.8423.2142.7412.2971.8153.1892.7212.1241.6423.0882.5240.890.890.900.890.900.910.890.900.900.910.900.91II2.2951.8013.1692.7022.2451.7653.1412.6712.0751.5903.0372.4800.900.900.900.890.910.910.900.910.910.910.900.90III2.2941.8183.1912.7222.2781.8013.1712.7022.1041.6193.0652.5030.890.900.900.890.900.910.900.900.900.910.910.91VI2.2881.8143.1872.7192.2691.8033.1642.7052.1011.6093.0512.4970.900.900.910.890.900.920.900.900.910.910.910.90(30, 30, 40, 1.2)I2.2661.7793.1412.6712.2141.7383.1142.6422.0441.5623.0112.4540.910.900.910.910.900.920.910.910.920.930.900.92II2.2191.7313.1022.6242.2701.7013.0752.6022.0011.5142.9502.4130.910.910.910.920.900.920.910.900.920.930.920.93III2.2451.7613.1192.6522.1911.7153.0912.6212.0251.5513.0022.4370.900.910.910.920.910.920.910.930.920.930.910.92VI2.2381.7623.1212.6482.1851.7093.0882.6182.0241.5473.0052.4290.910.910.930.920.910.940.920.930.960.930.920.94(50, 50, 60, 1.2)I2.1911.7033.0752.5912.2421.6713.0482.5701.9701.4752.9192.3850.920.900.930.920.950.920.910.920.920.930.940.96II2.1451.6503.0242.5442.2031.6243.0072.5191.9241.4292.8802.3270.920.920.930.940.950.920.930.920.930.930.940.92III2.1661.6773.0522.5642.2151.6453.0212.5421.9431.4492.9002.3610.930.920.930.920.940.920.900.920.920.940.940.94VI2.1571.6693.0482.5552.2171.6413.0172.5331.9451.4392.8922.3540.910.920.920.920.930.920.930.920.940.930.940.92(30, 30, 40, 2.2)I2.2471.7583.1252.6492.1851.7193.1002.6222.0251.5412.9802.4370.900.900.890.910.910.920.920.910.920.910.920.92II2.2021.7143.0832.6072.2511.6823.0552.5701.9831.4892.9312.3900.920.920.910.910.930.920.920.910.940.930.920.95III2.2241.7443.1022.6312.1691.7003.0742.6042.0031.5282.9842.4190.920.920.930.930.900.940.960.930.920.950.910.94VI2.2191.7453.1032.6312.1641.6913.0712.6022.0071.5322.9892.4110.920.940.930.900.910.940.920.930.940.920.930.93(50, 50, 60, 2.2)I2.1681.6793.0612.5682.2251.6493.0312.5481.9521.4612.9032.3710.920.900.940.920.930.920.910.930.920.960.940.94II2.1271.6293.0072.5262.1821.6072.9842.5031.9011.4082.8592.3010.910.940.930.920.950.940.950.920.950.930.950.94III2.1421.6613.0342.5412.2011.6283.0052.5191.9271.4312.8822.3390.920.920.940.920.940.920.960.920.970.940.930.93VI2.1381.6513.0372.5412.2081.6283.0082.5161.9191.4242.8812.3380.940.920.920.920.920.920.930.920.950.930.940.93Table 6
The average estimates and MSEs whenφ=0.5,0.3,0.6,0.2.
(D, N,m, τ)SCMLEBayes (P0)Bayes (P1)θ11θ12θ21θ22θ11θ12θ21θ22θ11θ12θ21θ22(30, 30, 30, 2.2)I0.6240.4120.7240.3250.6110.4030.7130.3140.5840.3750.6820.2750.2430.1750.2870.1170.2280.1690.2780.1120.1500.1020.2120.098II0.6090.4020.7080.3110.6020.3920.7040.3000.5710.3600.6690.2640.2150.1510.2540.0890.2020.1410.2510.0910.1240.0880.1830.085III0.6140.4090.7170.3150.6070.3950.7000.3020.5720.3710.6690.2680.2290.1620.2710.1080.2170.1540.2680.1030.1380.0910.2000.096VI0.6170.4020.7080.3070.6010.3960.7010.2940.5640.3610.6580.2540.2240.1550.2630.1010.2090.1480.2610.0970.1310.0840.1890.097(30, 30, 40, 2.2)I0.5890.3890.6910.2880.5810.3780.6840.2780.5620.3480.6510.2500.1950.1390.2380.0740.1890.1250.2350.0780.1090.0720.1690.073II0.5780.3720.6820.2690.5640.3600.6750.2610.5490.3690.6670.2350.1660.1120.2090.0510.1520.1010.2070.0510.0890.0410.1360.045III0.5810.3750.6730.2730.5690.3630.6650.2630.5510.3410.6350.2410.1820.1250.2220.0590.1740.1120.2190.0660.1000.0580.1520.060VI0.5770.3780.6780.2690.5580.3600.6570.2670.5540.3320.6280.2350.1740.1170.2150.0570.1630.1040.2150.0540.0910.0530.1480.049(50, 50, 60, 2.2)I0.5610.3490.6630.2510.5410.3380.6570.2390.5310.3440.6410.2190.1310.0910.1870.0420.1290.0800.1770.0400.0690.0350.1180.028II0.5470.3350.6480.2330.5270.3220.6430.2240.5200.3310.6280.2030.1140.0860.1690.0280.1140.0640.1600.0270.0440.0240.1030.015III0.5550.3420.6540.2480.5390.3320.6480.2280.5240.3370.6320.2120.1250.0840.1820.0370.1150.0770.1640.0280.0510.0240.1100.021VI0.5570.3380.6580.2330.5310.3240.6410.2190.5150.3280.6270.2070.1190.0790.1750.0290.1070.0640.1480.0190.0430.0190.1000.013(30, 30, 40, 3.2)I0.5780.3740.6830.2740.5700.3690.6710.2650.5490.3370.6390.2410.1780.1210.2190.0570.1710.1080.2140.0600.0890.0680.1450.059II0.5690.3610.6710.2560.5510.3470.6630.2480.5370.3560.6580.2220.1450.0940.1910.0670.1350.0870.1930.0380.0710.0290.1180.027III0.5720.3660.6620.2610.5580.3580.6540.2540.5470.3390.6240.2300.1640.1080.2040.0470.1560.1000.2040.0520.0850.0420.1380.045VI0.5690.3650.6640.2510.5510.3480.6450.2560.5430.3210.6190.2220.1570.1040.2010.0450.1480.0880.1950.0420.0800.0410.1290.031(50, 50, 60, 3.2)I0.5490.3380.6540.2390.5280.3250.6410.2300.5190.3320.6350.2080.1140.0740.1680.0270.1090.0640.1610.0290.0450.0220.1020.007II0.5270.3220.6390.2270.5180.3140.6290.2150.5080.3240.6170.1920.0910.0690.1510.0140.1000.0490.1430.0150.0250.0140.0820.011III0.5460.3370.6510.2400.5320.3240.6280.2210.5180.3310.6270.2040.1090.0710.1610.0190.1000.0610.1480.0150.0390.0150.0950.015VI0.5510.3310.6450.2280.5240.3150.6370.2130.5080.3170.6210.1920.1020.0620.1690.0140.890.0490.1370.0120.0310.0120.0840.009Table 7
The ALs and CPs of95% CIs when φ=0.5,0.3,0.6,0.2.
(D, N,m, τ)SCMLEBayes (P0)Bayes (P1)θ11θ12θ21θ22θ11θ12θ21θ22θ11θ12θ21θ22(30, 30, 30, 2.2)I1.2240.7411.4210.5211.2090.7251.4140.5071.1240.6241.3250.4420.880.890.890.880.890.900.890.900.900.910.900.91II1.1750.7011.2820.4791.1650.6271.3810.4611.0820.6741.2900.4030.900.890.900.900.910.920.900.910.900.910.940.92III1.2040.7271.4090.5041.1890.7111.4000.4921.1110.6051.3090.4270.890.890.900.900.890.900.900.900.900.910.920.90VI1.2110.7181.4020.5081.1770.7031.3920.4881.1040.6071.3130.41890.900.890.910.890.890.910.900.900.920.910.920.91(30, 30, 40, 2.2)I1.1510.6721.2560.4541.1410.6021.3560.4341.0570.6511.2660.3770.910.900.900.900.910.930.920.910.920.910.930.93II1.1140.6351.2180.4151.1070.5611.3210.4011.0210.6121.2280.3310.920.910.900.930.910.920.920.950.920.900.940.91III1.1330.6551.2380.4371.1240.5891.3390.4181.0410.6341.2450.3610.920.910.900.900.920.930.940.910.960.920.930.91VI1.1380.6491.2270.4281.1190.5911.3280.4091.0440.6271.2390.3650.900.910.900.910.920.930.920.910.960.900.930.93(50, 50, 60, 2.2)I1.0850.6111.1920.3891.0820.5351.3000.3751.0000.5871.2040.3070.930.920.900.930.930.920.940.930.930.900.940.93II1.0520.5771.1560.3531.0570.5011.2660.3410.9640.5531.1690.2890.930.940.940.930.930.920.930.930.970.920.940.94III1.0710.5921.1740.3711.0640.5181.2820.3590.9840.5711.1860.2890.900.930.920.930.930.930.920.930.940.920.910.95VI1.0760.5851.1670.3661.0570.5091.2770.34890.9790.5641.1790.2910.920.920.940.930.940.940.920.930.950.920.920.94(30, 30, 40, 3.2)I1.1340.6541.2390.4381.1270.5871.3380.4181.0410.6291.2480.3600.900.910.900.940.910.930.920.930.920.920.930.93II1.0900.6181.2030.4001.0920.5471.3050.3871.0080.6001.2110.3150.940.900.900.930.910.920.920.950.920.900.940.91III1.1330.6551.2380.4371.1240.5891.3390.4181.0410.6341.2450.3610.920.910.920.930.920.930.920.910.940.920.930.93VI1.1210.6311.2090.4131.1020.5811.3070.4001.0280.6141.2230.3510.920.920.940.930.920.930.930.910.960.920.930.92(50, 50, 60, 3.2)I1.0710.5921.1790.3741.0660.5191.2660.3610.9850.5691.1850.2920.920.910.900.910.930.920.920.930.950.920.940.95II1.0290.5581.1410.3341.0420.4821.2510.3270.9510.5371.1510.2710.920.920.940.920.930.930.920.910.900.940.940.96III1.0570.5771.1580.3491.0480.5071.2690.3390.9670.5591.1710.2710.950.930.920.910.940.930.920.940.940.930.920.94VI1.0610.5691.1540.34981.0380.4921.2620.34680.9640.5511.1540.2770.920.930.940.920.940.910.920.950.920.940.930.92


## 5.1. Data Analysis

Let us consider the real data sets given in Hoel [23]. The data given in Table 1 were obtained from a laboratory experiment on two groups of strain male mice which had received a radiation dose of 300r at an age of 5-6 weeks. The first group of mice (Group-1) lived in a conventional laboratory environment while the second group (Group-2) was in a germ-free environment. Two major causes of death are considered: thymic lymphoma and reticulum cell sarcoma for Cause 1 and Cause 2, respectively. Therefore, we consider that failures of the first group come from the Pop-η1 and the failures of the second group come from the Pop-η2. For more convenience, let us divide the data points by 1000, which will not affect the inference procedures. Under consideration that m=60, τ=0.380, and R=0,0,…,0,68, the jointly Type-I HPCS competing risks random sample is presented in Table 2. The data reported in Table 2 show that J=56,m1,m2=33,23 and n11,n12,n21,n22=17,16,19,4. The random combined sample is selected from both populations, Pop-η1 and Pop-η2, of sizes D=61 and N=67, respectively. For Bayesian analysis, since we do not have any prior information about the model parameters, we consider the noninformative prior by setting ai=bi=0.0001,i=1,2,3,4. Hence, the MLEs and BEs under SE, LINEX, and GE losses and the corresponding CIs are computed and presented in Table 3. It can be observed from Table 3 that all point estimates are lying within the 95% asymptotic CIs and Bayes CrIs.Table 1
Time to failure of male mice which received a radiation dose at age 5-6 weeks.
Group 1Thymic lymphoma159189191198200207220235245250256261265266280343356383403414428432Other cases40425162163179206222228249252282324333341366385407420431441461462482517517524564567586619620621622647651686761763Group 2Thymic lymphoma158192193194195202212215229230237240244247259300301321337415434444485496529537624707800Other cases13624625537642156561661765265565866066267568173473673775776977780080782585585786486887087388289591093494210151019Table 2
The Type-II HPC competing risks sample for radiation dose data withm=60.
ωi11112211111222ti0.0400.0420.0510.0620.1360.1580.1590.1630.1790.1890.1910.1920.1930.194δi00000110011111ωi21121122111221ti0.1950.1980.2000.2020.2060.2070.2120.2150.2200.2220.2280.2290.2300.235δi11110111100111ωi22212211121211ti0.2370.2400.2440.2450.2460.2470.2490.2500.2520.2550.2560.2590.2610.265δi11110101001111ωi11122211211112ti0.2660.2800.2820.3000.3010.3210.3240.3330.3370.3410.3430.3560.3660.376δi11011100101100Table 3
The MLEs and BEs of the parameters and corresponding CIs.
MLSELLINXGE95% asymptotic CI95% Bayes CrIsc = 1.0c = −1.0q = 2.0q = −2.0θ110.93920.93920.91420.96610.85590.9664(0.4927, 1.3856)(0.5984, 1.3425)θ120.88390.88390.86040.90930.80060.9111(0.4508, 1.3170)(0.5544, 1.2760)θ210.85440.85450.83570.87420.78660.8766(0.4702, 1.2385)(0.5595, 1.2002)θ220.17990.17990.17590.18400.11020.2011(0.0036, 0.3561)(0.0614, 0.3487)


## 5.2. Simulation Study

Now, we compare the performances of the different methods of estimation based on Monte Carlo simulations. We compare the performances of the MLEs and BEs in terms of biases and mean square errors (MSEs). In this simulation, two sets of parameter values,φ=1.0,0.8,1.5,1.2 and 0.5,0.3,0.6,0.2, are considered. Here, we consider different effective sample sizes m=30,40,60 and different censoring schemes. In this study, we check how the effect of changing the sample size D+N,m censoring scheme R and the ideal test time τ is. Under the noninformative and informative priors, the biases and MSEs of the estimates over 1000 replications are computed for various censoring schemes. The algorithm used for the numerical computations can be described as follows:(1)
For given prior informationai,bii=1,2,3,4, the true value of the parameter vector φ=θ11,θ12,θ21,θ22 is selected almost to satisfy φi=Eφi=ai/bi.(2)
For givenR, the value R1i is generated from Ri binomial variant with the probability of success being equal to 1/2 and R2i=Ri−R1i.(3)
For givenm, generate m1 from m binomial distribution with probability of success being equal to 1/2. Then, m2=m−m1.(4)
Generate Type-I HPCS competing risks samples of sizemk from exponential distributions with parameter θk1+θk2,k=1,2.(5)
For givenτ and the joint Type-I HPC competing risks sample, determine the joint hybrid progressive Type-I competing risks sample.(6)
From the joint Type-I progressive competing risks, censoring sample determinesm1 and m2.(7)
The discrete random variablesnkj,k,j=1,2, are generated from binomial distributions with sample sizes m1 and m2 with the probability of success θk1/θk1+θk2 and θk2/θk1+θk2, respectively.(8)
The point and interval estimates based on the likelihood and Bayes approaches are computed.(9)
For comparison purposes, the average estimates (AEs) and MSEs are computed. Furthermore, the average lengths (ALs) and coverage probabilities (CPs) are evaluated.(10)
The simulation study is conducted using censoring schemes (CSs) described as follows:(38)CS=I:Ri=0,0,…,0,D+N−m,II:Ri=D+N−m,0,0,…,0,III:Ri=0,i≠m+12andmodd ori≠m2andmeven,D+N−m,i=m+12andmodd ori=m2andmeven,IV:Ri=0,i>D+N−m2,2,i≤D+N−m2.(11)
All results of the simulation study are reported in Tables4–7 for noninformative prior P0=ai,bii=1,2,3,4=0.0001,0.0001 and informative prior P1.Table 4
The average estimates and MSEs whenφ=1.0,0.8,1.5,1.2.
(D, N,m, τ)SCMLEBayes (P0)Bayes (P1)θ11θ12θ21θ22θ11θ12θ21θ22θ11θ12θ21θ22(30, 30, 30, 1.2)I1.2311.1241.6211.4251.2111.1121.6071.4091.1870.9821.58721.3140.2410.2040.4210.3740.2170.1920.4030.3550.1890.1140.3540.301II1.1991.1111.6041.4131.1871.0951.5891.3391.1420.9411.58391.2870.1920.1520.3720.3270.1780.1510.3490.3040.1410.0970.3040.251III1.2161.1071.6141.4131.2021.1031.5931.4011.1680.9711.5861.3030.2240.1900.4040.3550.2000.1750.3840.3370.1710.1010.3360.289VI1.2071.1111.6171.4081.2051.1091.5891.4041.1540.9591.5811.2980.2130.1970.4010.3420.1920.1640.3790.3310.1660.1090.3280.281(30, 30, 40, 1.2)I1.1321.0451.5871.3561.1221.0221.5141.3011.0890.9021.5271.2290.1670.1180.3480.3010.1480.1230.3180.2880.1150.0690.2890.214II1.1191.0101.5651.3391.1041.0121.5101.2891.0660.8891.5011.2070.1140.0850.3010.2510.1000.0800.2790.2420.0890.0540.2390.179III1.1191.0391.5811.3441.1181.0171.5071.3031.0810.8971.5191.2210.1490.1040.3310.2910.1320.1070.3040.2790.1000.0610.2790.211VI1.1211.0321.5781.3391.1211.0241.5131.3081.0780.8821.5081.2150.1380.1010.3240.2840.1190.1040.3080.2660.0940.0630.2810.198(50, 50, 60, 1.2)I1.1040.9821.5421.3141.1100.9781.5291.3031.0610.8751.5111.2090.0890.0610.2420.2040.0870.0740.2520.2140.0830.0540.2100.129II1.0990.9791.5281.3141.1070.9691.5111.3121.0580.8711.5031.2040.0720.0540.2090.1870.0820.0690.2250.1820.0710.0520.2070.114III1.1110.9791.5331.3101.1120.9711.5241.3111.0580.8711.5141.2120.0840.0580.2150.2060.0810.0710.2480.2090.0810.0550.2100.114VI1.1130.9721.5301.3071.1150.9821.5211.3081.0530.8661.5021.2040.0810.0540.2120.2080.0770.0750.2420.2030.0750.0510.2070.105(30, 30, 40, 2.2)I1.1121.0241.5651.3331.1011.0021.5011.2891.0700.8821.5111.2120.1450.1010.3220.2820.1250.1020.3000.2650.1010.0480.2650.191II1.1111.0011.5571.3251.0821.0071.5031.2781.0580.8721.4901.1890.0920.0960.2850.2360.0890.0760.2630.2310.0800.0510.2280.165III1.1201.0291.5751.3401.1071.0111.4971.2881.0640.8831.5011.2030.1240.0810.3040.2690.10110.0890.2810.2520.0830.0470.2640.190VI1.1051.0211.5491.3271.1091.0011.4951.2891.0560.8541.2891.2030.1190.0910.3010.2640.0870.0940.3000.2490.0880.0590.2620.182(50, 50, 60, 2.2)I1.0890.9751.5291.3011.1030.9641.5141.2851.0450.8651.5031.2010.0890.0610.2420.2040.0870.0740.2520.2140.0830.0540.2100.129II1.0810.9681.5191.3031.1020.9481.5011.3041.0510.8591.5101.2010.0680.0490.1880.1550.0710.0610.2110.1690.0640.0490.2010.101III1.1030.9681.5251.3041.1030.9641.5131.3021.0510.8681.5011.2030.0750.0510.2000.1880.0720.0610.2190.1870.0730.0520.2020.103VI1.1090.9651.5241.3021.1040.9771.5141.3031.0440.8581.4921.2030.0740.0470.2000.1890.0620.0640.2240.1810.0630.0490.2010.099Table 5
The ALs and CPs of95% CIs when φ=1.0,0.8,1.5,1.2.
(D, N,m, τ)SCMLEBayes (P0)Bayes (P1)θ11θ12θ21θ22θ11θ12θ21θ22θ11θ12θ21θ22(30, 30, 30, 1.2)I2.3151.8423.2142.7412.2971.8153.1892.7212.1241.6423.0882.5240.890.890.900.890.900.910.890.900.900.910.900.91II2.2951.8013.1692.7022.2451.7653.1412.6712.0751.5903.0372.4800.900.900.900.890.910.910.900.910.910.910.900.90III2.2941.8183.1912.7222.2781.8013.1712.7022.1041.6193.0652.5030.890.900.900.890.900.910.900.900.900.910.910.91VI2.2881.8143.1872.7192.2691.8033.1642.7052.1011.6093.0512.4970.900.900.910.890.900.920.900.900.910.910.910.90(30, 30, 40, 1.2)I2.2661.7793.1412.6712.2141.7383.1142.6422.0441.5623.0112.4540.910.900.910.910.900.920.910.910.920.930.900.92II2.2191.7313.1022.6242.2701.7013.0752.6022.0011.5142.9502.4130.910.910.910.920.900.920.910.900.920.930.920.93III2.2451.7613.1192.6522.1911.7153.0912.6212.0251.5513.0022.4370.900.910.910.920.910.920.910.930.920.930.910.92VI2.2381.7623.1212.6482.1851.7093.0882.6182.0241.5473.0052.4290.910.910.930.920.910.940.920.930.960.930.920.94(50, 50, 60, 1.2)I2.1911.7033.0752.5912.2421.6713.0482.5701.9701.4752.9192.3850.920.900.930.920.950.920.910.920.920.930.940.96II2.1451.6503.0242.5442.2031.6243.0072.5191.9241.4292.8802.3270.920.920.930.940.950.920.930.920.930.930.940.92III2.1661.6773.0522.5642.2151.6453.0212.5421.9431.4492.9002.3610.930.920.930.920.940.920.900.920.920.940.940.94VI2.1571.6693.0482.5552.2171.6413.0172.5331.9451.4392.8922.3540.910.920.920.920.930.920.930.920.940.930.940.92(30, 30, 40, 2.2)I2.2471.7583.1252.6492.1851.7193.1002.6222.0251.5412.9802.4370.900.900.890.910.910.920.920.910.920.910.920.92II2.2021.7143.0832.6072.2511.6823.0552.5701.9831.4892.9312.3900.920.920.910.910.930.920.920.910.940.930.920.95III2.2241.7443.1022.6312.1691.7003.0742.6042.0031.5282.9842.4190.920.920.930.930.900.940.960.930.920.950.910.94VI2.2191.7453.1032.6312.1641.6913.0712.6022.0071.5322.9892.4110.920.940.930.900.910.940.920.930.940.920.930.93(50, 50, 60, 2.2)I2.1681.6793.0612.5682.2251.6493.0312.5481.9521.4612.9032.3710.920.900.940.920.930.920.910.930.920.960.940.94II2.1271.6293.0072.5262.1821.6072.9842.5031.9011.4082.8592.3010.910.940.930.920.950.940.950.920.950.930.950.94III2.1421.6613.0342.5412.2011.6283.0052.5191.9271.4312.8822.3390.920.920.940.920.940.920.960.920.970.940.930.93VI2.1381.6513.0372.5412.2081.6283.0082.5161.9191.4242.8812.3380.940.920.920.920.920.920.930.920.950.930.940.93Table 6
The average estimates and MSEs whenφ=0.5,0.3,0.6,0.2.
(D, N,m, τ)SCMLEBayes (P0)Bayes (P1)θ11θ12θ21θ22θ11θ12θ21θ22θ11θ12θ21θ22(30, 30, 30, 2.2)I0.6240.4120.7240.3250.6110.4030.7130.3140.5840.3750.6820.2750.2430.1750.2870.1170.2280.1690.2780.1120.1500.1020.2120.098II0.6090.4020.7080.3110.6020.3920.7040.3000.5710.3600.6690.2640.2150.1510.2540.0890.2020.1410.2510.0910.1240.0880.1830.085III0.6140.4090.7170.3150.6070.3950.7000.3020.5720.3710.6690.2680.2290.1620.2710.1080.2170.1540.2680.1030.1380.0910.2000.096VI0.6170.4020.7080.3070.6010.3960.7010.2940.5640.3610.6580.2540.2240.1550.2630.1010.2090.1480.2610.0970.1310.0840.1890.097(30, 30, 40, 2.2)I0.5890.3890.6910.2880.5810.3780.6840.2780.5620.3480.6510.2500.1950.1390.2380.0740.1890.1250.2350.0780.1090.0720.1690.073II0.5780.3720.6820.2690.5640.3600.6750.2610.5490.3690.6670.2350.1660.1120.2090.0510.1520.1010.2070.0510.0890.0410.1360.045III0.5810.3750.6730.2730.5690.3630.6650.2630.5510.3410.6350.2410.1820.1250.2220.0590.1740.1120.2190.0660.1000.0580.1520.060VI0.5770.3780.6780.2690.5580.3600.6570.2670.5540.3320.6280.2350.1740.1170.2150.0570.1630.1040.2150.0540.0910.0530.1480.049(50, 50, 60, 2.2)I0.5610.3490.6630.2510.5410.3380.6570.2390.5310.3440.6410.2190.1310.0910.1870.0420.1290.0800.1770.0400.0690.0350.1180.028II0.5470.3350.6480.2330.5270.3220.6430.2240.5200.3310.6280.2030.1140.0860.1690.0280.1140.0640.1600.0270.0440.0240.1030.015III0.5550.3420.6540.2480.5390.3320.6480.2280.5240.3370.6320.2120.1250.0840.1820.0370.1150.0770.1640.0280.0510.0240.1100.021VI0.5570.3380.6580.2330.5310.3240.6410.2190.5150.3280.6270.2070.1190.0790.1750.0290.1070.0640.1480.0190.0430.0190.1000.013(30, 30, 40, 3.2)I0.5780.3740.6830.2740.5700.3690.6710.2650.5490.3370.6390.2410.1780.1210.2190.0570.1710.1080.2140.0600.0890.0680.1450.059II0.5690.3610.6710.2560.5510.3470.6630.2480.5370.3560.6580.2220.1450.0940.1910.0670.1350.0870.1930.0380.0710.0290.1180.027III0.5720.3660.6620.2610.5580.3580.6540.2540.5470.3390.6240.2300.1640.1080.2040.0470.1560.1000.2040.0520.0850.0420.1380.045VI0.5690.3650.6640.2510.5510.3480.6450.2560.5430.3210.6190.2220.1570.1040.2010.0450.1480.0880.1950.0420.0800.0410.1290.031(50, 50, 60, 3.2)I0.5490.3380.6540.2390.5280.3250.6410.2300.5190.3320.6350.2080.1140.0740.1680.0270.1090.0640.1610.0290.0450.0220.1020.007II0.5270.3220.6390.2270.5180.3140.6290.2150.5080.3240.6170.1920.0910.0690.1510.0140.1000.0490.1430.0150.0250.0140.0820.011III0.5460.3370.6510.2400.5320.3240.6280.2210.5180.3310.6270.2040.1090.0710.1610.0190.1000.0610.1480.0150.0390.0150.0950.015VI0.5510.3310.6450.2280.5240.3150.6370.2130.5080.3170.6210.1920.1020.0620.1690.0140.890.0490.1370.0120.0310.0120.0840.009Table 7
The ALs and CPs of95% CIs when φ=0.5,0.3,0.6,0.2.
(D, N,m, τ)SCMLEBayes (P0)Bayes (P1)θ11θ12θ21θ22θ11θ12θ21θ22θ11θ12θ21θ22(30, 30, 30, 2.2)I1.2240.7411.4210.5211.2090.7251.4140.5071.1240.6241.3250.4420.880.890.890.880.890.900.890.900.900.910.900.91II1.1750.7011.2820.4791.1650.6271.3810.4611.0820.6741.2900.4030.900.890.900.900.910.920.900.910.900.910.940.92III1.2040.7271.4090.5041.1890.7111.4000.4921.1110.6051.3090.4270.890.890.900.900.890.900.900.900.900.910.920.90VI1.2110.7181.4020.5081.1770.7031.3920.4881.1040.6071.3130.41890.900.890.910.890.890.910.900.900.920.910.920.91(30, 30, 40, 2.2)I1.1510.6721.2560.4541.1410.6021.3560.4341.0570.6511.2660.3770.910.900.900.900.910.930.920.910.920.910.930.93II1.1140.6351.2180.4151.1070.5611.3210.4011.0210.6121.2280.3310.920.910.900.930.910.920.920.950.920.900.940.91III1.1330.6551.2380.4371.1240.5891.3390.4181.0410.6341.2450.3610.920.910.900.900.920.930.940.910.960.920.930.91VI1.1380.6491.2270.4281.1190.5911.3280.4091.0440.6271.2390.3650.900.910.900.910.920.930.920.910.960.900.930.93(50, 50, 60, 2.2)I1.0850.6111.1920.3891.0820.5351.3000.3751.0000.5871.2040.3070.930.920.900.930.930.920.940.930.930.900.940.93II1.0520.5771.1560.3531.0570.5011.2660.3410.9640.5531.1690.2890.930.940.940.930.930.920.930.930.970.920.940.94III1.0710.5921.1740.3711.0640.5181.2820.3590.9840.5711.1860.2890.900.930.920.930.930.930.920.930.940.920.910.95VI1.0760.5851.1670.3661.0570.5091.2770.34890.9790.5641.1790.2910.920.920.940.930.940.940.920.930.950.920.920.94(30, 30, 40, 3.2)I1.1340.6541.2390.4381.1270.5871.3380.4181.0410.6291.2480.3600.900.910.900.940.910.930.920.930.920.920.930.93II1.0900.6181.2030.4001.0920.5471.3050.3871.0080.6001.2110.3150.940.900.900.930.910.920.920.950.920.900.940.91III1.1330.6551.2380.4371.1240.5891.3390.4181.0410.6341.2450.3610.920.910.920.930.920.930.920.910.940.920.930.93VI1.1210.6311.2090.4131.1020.5811.3070.4001.0280.6141.2230.3510.920.920.940.930.920.930.930.910.960.920.930.92(50, 50, 60, 3.2)I1.0710.5921.1790.3741.0660.5191.2660.3610.9850.5691.1850.2920.920.910.900.910.930.920.920.930.950.920.940.95II1.0290.5581.1410.3341.0420.4821.2510.3270.9510.5371.1510.2710.920.920.940.920.930.930.920.910.900.940.940.96III1.0570.5771.1580.3491.0480.5071.2690.3390.9670.5591.1710.2710.950.930.920.910.940.930.920.940.940.930.920.94VI1.0610.5691.1540.34981.0380.4921.2620.34680.9640.5511.1540.2770.920.930.940.920.940.910.920.950.920.940.930.92


## 6. Conclusions and Remarks

In this paper, we consider the problem of estimating the unknown parameters based on the joint Type-I HPC sample coming from two exponential distributions under a competing risks model. The likelihood and Bayesian methods are used for estimating the model parameter involved. In the Bayesian context, the Bayes estimates are obtained under different loss error functions. The asymptotic confidence intervals and Bayes credible intervals are also discussed. It is evident from the numerical results that while the MLEs are close to the ones based on the Bayes method under noninformative prior, the Bayes estimators under informative prior perform well when compared to the MLEs and the Bayes estimates under the noninformative prior. It is observed that when the effective samplem increases, the MSEs of the estimates and the ALs of the confidence intervals get to be decreased. Moreover, the MSEs of the point estimates as well as the ALs and CPs of CIs and CrIs improve when τ gets large. This observation is expected due to the fact that when τ increases, more information on the data is obtained.



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*Source: 3380467-2021-08-25.xml*