# Statistical Analysis for Competing Risks’ Model with Two Dependent Failure Modes from Marshall–Olkin Bivariate Gompertz Distribution

Authors: Min Wu; Fode Zhang; Yimin Shi; Yan Wang
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

The bivariate or multivariate distribution can be used to account for the dependence structure between different failure modes. This paper considers two dependent competing failure modes from Gompertz distribution, and the dependence structure of these two failure modes is handled by the Marshall–Olkin bivariate distribution. We obtain the maximum likelihood estimates (MLEs) based on classical likelihood theory and the associated bootstrap confidence intervals (CIs). The posterior density function based on the conjugate prior and noninformative (Jeffreys and Reference) priors are studied; we obtain the Bayesian estimates in explicit forms and construct the associated highest posterior density (HPD) CIs. The performance of the proposed methods is assessed by numerical illustration.


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## Body

## 1. Introduction

It is extremely common that the failure of a product or a system contains several competing failure modes in reliability engineering; any failure mode will lead to the failure result. Competing risks’ data contain the failure time and the corresponding failure mode, which can be modeled by the competing risks’ model and has been commonly performed in many research fields, such as engineering and medical statistics. Previous studies have mostly assumed the competing failure modes to be independent; Wang et al. [1], Ren and Gui [2], and Qin and Gui [3] focused on the independent competing risks’ model under progressively hybrid censoring from Weibull and Burr-XII distributions. Objective Bayesian analysis for the competing risks’ model with Wiener degradation phenomena and catastrophic failures was studied by Guan et al. [4]. In practice, the independency relationship between different failure modes is a very special case; a more common situation is dependency. That is, the failure mechanisms are interactive and interdependent; the occurrence of one failure mode will affect the occurrence of other failure modes. For example, a ship fixed carbon dioxide fire extinguishing system can fail due to pressure gauge, distribution value, cylinder group, and so on; these failure modes are dependent because they all are related to the storage environment. Therefore, it is more reasonable to assume dependency among different competing failure modes. The competing risks’ model considers the product or system with multiple dependent competing failure modes, any one of which will cause the occurrence of failure. The dependent competing risks’ model has been extensively studied. Zhang et al. [5] and Zhang et al. [6] studied the dependent competing risks’ model under accelerated life testing (ALT) by copula function to measure the dependence between different competing failure modes; the results indicate the copula construction method has good accuracy and universality. Wang and Yan [7] and Wu et al. [8] also studied this model under ALT and progressively hybrid-censoring scheme using Clayton copula and Gumbel copula, respectively. For other related works, see the works of Lo and Wike [9] and Fang et al. [10].In addition to using copula function to handle the relationship between different competing failure modes, the bivariate or multivariate distribution also can be used to account for the correlation between different failure modes. The Marshall–Olkin distribution [11], which has many good properties, is the best-known bivariate distribution and has been discussed extensively; it has a parameter to describe the dependence structure. Li et al. [12], Kundu and Gupta [13], and Bai et al. [14] provided reviews on Marshall–Olkin–Weibull distribution; Kundu and Gupta [13] obtained the explicit forms of the unknown parameters when the shape parameter is known; when the shape parameter is unknown, they used the importance sampling to compute the Bayesian estimates of the unknown parameters. Bai et al. [14] discussed the statistical analysis for the accelerated dependent competing risks’ model under Type-II hybrid censoring schemes. Guan et al. [15] studied objective Bayesian analysis for the Marshall–Olkin exponential distribution based on reference priors; they also found that some of the reference priors are also matching priors and the posterior distributions based on these priors are proper.The Gompertz distribution is a widely used growth model which has been studied extensively; Ismail [16] studied the Bayesian analysis of Gompertz distribution parameters and acceleration factor in the case of partially accelerated life testing under Type-I censoring scheme. Ghitany et al. [17] considered a progressively censored sample from Gompertz distribution; they discussed the existence and uniqueness of the MLEs of the unknown parameters. The Gompertz distribution plays an important role in fitting clinical trials’ data in medical science and can be used to the theory of extreme-order statistics. In this paper, we will study the dependent competing risks’ model from the Marshall–Olkin bivariate Gompertz (MOGP) distribution, which is a bivariate distribution with Gompertz marginal distributions. We focus our attention on the statistical analysis of the model parameters, including classical likelihood inference, Bayesian analysis, and objective Bayesian analysis. Because the Bayesian analysis based on conjugate prior is sensitive to the hyperparameters, inappropriate choice of it will cause bad priors. Based on this reason, we propose the objective Bayesian analysis based on noninformative priors for comparison. The objective Bayesian inference has been studied by Guan et al. [14], Bernardo [18], and Berger and Bernardo [19] based on Reference and Jeffreys priors.In the rest of this paper, we will present the model description and some properties. Section3 presents the MLEs and associated bootstrap CIs. In Section 4, Bayesian estimates and associated HPD CIs based on conjugate prior, Jeffreys prior [20], and reference priors [18] are obtained, and these priors lead to proper posteriors which are proved. Section 5 presents some results obtained from simulation study and illustrative analysis. Section 6 gives some final concluding remarks.


## 2. Model Description

Suppose thatft;λ,θ is a Gompertz distribution; the density function and reliability function of it are(1)ft;λ,θ=θeλt−θeλt−1/λ,λ,θ>0,t>0,St;λ,θ=e−θeλt−1/λ,λ,θ>0,t>0,where λ is shape parameter and θ is scale parameter.SupposeU0,U1,andU2 are three independent Gompertz variables with different scale parameters, that is, U0∼GPλ,θ0, U1∼GPλ,θ1, and U2∼GPλ,θ2. Let T1=minU0,U1 and T2=minU0,U2; we obtain T1∼GPλ,θ0+θ1 and T2∼GPλ,θ0+θ2. Then, the pair of variables T1,T2 follows the MOGP distribution denoted by T1,T2∼MOGPλ,θ0,θ1,θ2. When θ0=0, the two variables T1andT2 are independent and T1andT2 will be dependent when θ0>0; hence, θ0 can be regarded as a correlation coefficient between T1andT2.The joint PDF ofT1,T2 can be written as(2)fT1,T2t1,t2;λ,θ0,θ1,θ2=ft1;λ,θ0+θ1ft2;λ,θ2t1>t2ft1;λ,θ1ft2;λ,θ0+θ2t1<t2.θ0θ0+θ1+θ2ft;λ,θ0+θ1+θ2t1=t2=t.The surface plots offT1,T2t1,t2;λ,θ0,θ1,θ2 are presented in Figure 1. From Figure 1, we can see that fT1,T2t1,t2;λ,θ0,θ1,θ2 is a unimodal function.Figure 1
Surface plot offT1,T2t1,t2;λ,θ0,θ1,θ2 with different values of λ,θ0,θ1,θ2. (a) λ,θ0,θ1,θ2=3,0.5,2,1. (b) λ,θ0,θ1,θ2=3,1.5,0.5,2. (c) λ,θ0,θ1,θ2=1,0.5,0.5,0.5. (d) λ,θ0,θ1,θ2=1,0.2,0.8,0.6.
(a)(b)(c)(d)Putn identical products into test, and each product has two dependent failure modes with lifetimes T1,T2, T1,T2∼MOGPλ,θ0,θ1,θ2. Then, the system lifetime is X=minT1,T2∼MOGPλ,θ0+θ1+θ2. Let δ0l=IT1l=T2l, δ1l=IT1l<T2l, and δ2l=IT1l>T2l, for l=1,⋯,n, where I⋅ is an indicator function. Then, we can compute n0=∑lδ0l, n1=∑lδ1l,n2=∑lδ2l, and n=n0+n1+n2.Theorem 1.
Forl=1,⋯,n, δ0l=IT1l=T2l, δ1l=IT1l<T2l, and δ2l=IT1l>T2l, We have(3)δ0l,δ1l,δ2l∼Multinomial1;θ0θ0+θ1+θ2,θ1θ0+θ1+θ2,θ2θ0+θ1+θ2,l=1,⋯,n.Proof.
Forl=1,⋯,n, we have δ0l+δ1l+δ2l=1,(4)PT1<T2=∫0∞∫0t2ft1;λ,θ1ft2;λ,θ0+θ2dt1dt2=θ1θ0+θ1+θ2,PT1>T2=∫0∞∫0t1ft1;λ,θ0+θ1ft2;λ,θ2dt2dt1=θ2θ0+θ1+θ2,PT1=T2=1−θ1θ0+θ1+θ2−θ2θ0+θ1+θ2=θ0θ0+θ1+θ2.
Therefore,δ0l,δ1l,δ2l∼Multinomial1;θ0/θ0+θ1+θ2,θ1/θ0+θ1+θ2,θ2/θ0+θ1+θ2.
The likelihood function is(5)Lλ,θ0,θ1,θ2=∏lfT1,T2xl,xl;λ,θ0,θ1,θ2δ0l−∂ST1,T2t1,t2;λ,θ0,θ1,θ2∂t1|xl,xlδ1l,×−∂ST1,T2t1,t2;λ,θ0,θ1,θ2∂t2|xl,xlδ2l,where(6)fT1,T2xl,xl;λ,θ0,θ1,θ2=θ0θ0+θ1+θ2ft;λ,θ0+θ1+θ2,=θ0expλxl−θ0+θ1+θ2λeλxl−1,−∂ST1,T2t1,t2;λ,θ0,θ1,θ2∂t1|xl,xl,=θ1expλxl−θ0+θ1+θ2λeλxl−1,−∂ST1,T2t1,t2;λ,θ0,θ1,θ2∂t2|xl,xl,=θ2expλxl−θ0+θ1+θ2λeλxl−1.
Then, we obtain(7)Lx;λ,θ0,θ1,θ2=θ0n0θ1n1θ2n2expλ∑lxl−θ0+θ1+θ2λ∑leλxl−1.


## 3. Classical Inference

### 3.1. Maximum Likelihood Estimates (MLEs)

The MLEs ofθ0,θ1,θ2,andλ can be obtained by maximizing the logarithm of Lx;λ,θ0,θ1,θ2. Set the first partial derivation of logLx;λ,θ0,θ1,θ2 about θ0,θ1,θ2,λ to 0, i.e.,(8)∂logLx;λ,θ0,θ1,θ2∂λ=∑lxl+θ0+θ1+θ2λ2∑leλxl−1−θ0+θ1+θ2λ∑lxleλxl=0,∂logLx;λ,θ0,θ1,θ2∂θj=njθj−1λ∑leλxl−1=0,j=0,1,2.From (8), we get the MLEs of θ0,θ1,andθ2 as(9)θ^jλ=njλ∑leλxl−1,j=0,1,2.Substitutingθ^jλ into logLx;λ,θ0,θ1,θ2, we obtain(10)hλ=λ∑lxl+∑j=02lnnjλ∑leλxl−1nj,which is the profile logarithm likelihood function of λ.We can show that∂2hλ/∂λ2<0, which implies that hλ is concave. Some iterative schemes can be used to find the MLE for λ, such as Newton–Raphson algorithm.


### 3.2. Bootstrap Confidence Intervals (CIs)

Since it is hard to construct the exact CIs for the unknown parameters, we consider the Bootstrap method to construct CIs for parametersθ0,θ1,θ2,andλ. The Bootstrap method is a resampling method to estimate some statistical characteristics for the unknown parameters by taking samples from the original samples repeatedly; the obtained samples are called Bootstrap samples. This method has a great practical value since it does not need to assume the overall distribution or construct the pivot quantity. We generate the Bootstrap sample by the following three steps:Step 1: for the fixed value ofn and observed data x1,x2,⋯,xn, we get the estimates λ^,θ^0,θ^1,andθ^2 based on the maximum likelihood method.Step 2: for the values ofn, λ^,θ^0,θ^1,andθ^2, we generate the sample x1∗,x2∗,⋯,xn∗. Then, get the MLEs λ^'′,θ^0'′,θ^1'′,andθ^2'′.Step 3: repeat Step 2M times to obtain M sets of the values λ^'′,θ^0'′,θ^1'′,andθ^2'′. Arrange them as follows to get the Bootstrap sample:(11)θ^01'′<⋯<θ^0M'′,θ^11'′<⋯<θ^1M'′,θ^21'′<⋯<θ^2M'′,λ^1'′<⋯<λ^M'′.Based on the Bootstrap sample and by percentile Bootstrap (Boot-P) method, we construct the Boot-P CIs forθ0,θ1,θ2,λ at 1−γ confidence level as(12)θ^0Mγ/2″,θ^0M1−γ/2″,θ^1Mγ/2″,θ^1M1−γ/2″,θ^2Mγ/2″,θ^2M1−γ/2″,λMγ/2″,λM1−γ/2″.


## 3.1. Maximum Likelihood Estimates (MLEs)

The MLEs ofθ0,θ1,θ2,andλ can be obtained by maximizing the logarithm of Lx;λ,θ0,θ1,θ2. Set the first partial derivation of logLx;λ,θ0,θ1,θ2 about θ0,θ1,θ2,λ to 0, i.e.,(8)∂logLx;λ,θ0,θ1,θ2∂λ=∑lxl+θ0+θ1+θ2λ2∑leλxl−1−θ0+θ1+θ2λ∑lxleλxl=0,∂logLx;λ,θ0,θ1,θ2∂θj=njθj−1λ∑leλxl−1=0,j=0,1,2.From (8), we get the MLEs of θ0,θ1,andθ2 as(9)θ^jλ=njλ∑leλxl−1,j=0,1,2.Substitutingθ^jλ into logLx;λ,θ0,θ1,θ2, we obtain(10)hλ=λ∑lxl+∑j=02lnnjλ∑leλxl−1nj,which is the profile logarithm likelihood function of λ.We can show that∂2hλ/∂λ2<0, which implies that hλ is concave. Some iterative schemes can be used to find the MLE for λ, such as Newton–Raphson algorithm.


## 3.2. Bootstrap Confidence Intervals (CIs)

Since it is hard to construct the exact CIs for the unknown parameters, we consider the Bootstrap method to construct CIs for parametersθ0,θ1,θ2,andλ. The Bootstrap method is a resampling method to estimate some statistical characteristics for the unknown parameters by taking samples from the original samples repeatedly; the obtained samples are called Bootstrap samples. This method has a great practical value since it does not need to assume the overall distribution or construct the pivot quantity. We generate the Bootstrap sample by the following three steps:Step 1: for the fixed value ofn and observed data x1,x2,⋯,xn, we get the estimates λ^,θ^0,θ^1,andθ^2 based on the maximum likelihood method.Step 2: for the values ofn, λ^,θ^0,θ^1,andθ^2, we generate the sample x1∗,x2∗,⋯,xn∗. Then, get the MLEs λ^'′,θ^0'′,θ^1'′,andθ^2'′.Step 3: repeat Step 2M times to obtain M sets of the values λ^'′,θ^0'′,θ^1'′,andθ^2'′. Arrange them as follows to get the Bootstrap sample:(11)θ^01'′<⋯<θ^0M'′,θ^11'′<⋯<θ^1M'′,θ^21'′<⋯<θ^2M'′,λ^1'′<⋯<λ^M'′.Based on the Bootstrap sample and by percentile Bootstrap (Boot-P) method, we construct the Boot-P CIs forθ0,θ1,θ2,λ at 1−γ confidence level as(12)θ^0Mγ/2″,θ^0M1−γ/2″,θ^1Mγ/2″,θ^1M1−γ/2″,θ^2Mγ/2″,θ^2M1−γ/2″,λMγ/2″,λM1−γ/2″.


## 4. Bayesian Inference and HPD CIs

### 4.1. Conjugate Prior

In this section, we suppose the shape parameterλ is known. Denote θ=θ0+θ1+θ2, which has a Gamma prior with hyperparameters a and b as(13)πθ=baΓaθa−1e−bθ,a>0,b>0,θ>0.Due toθ0/θ+θ1/θ+θ2/θ=1, so given θ, θ1/θ,θ2/θ follows a Dirichlet prior with hyper parameters c0,c1, and c2, that is,(14)πDθ1θ,θ2θ|θ=Γ∑i=02ci∏i=02Γci∏i=02θiθci−1,θi>0,ci>0,i=0,1,2.Therefore, the joint prior ofθ0,θ1,andθ2 becomes(15)π1θ0,θ1,θ2;a,b,c0,c1,c2=ΓcΓabθa−c∏i=02bciΓciθici−1exp−bθi,where c=c0+c1+c2.


### 4.2. Jeffreys Prior

According to Jeffreys [20], Jeffreys prior is proportional to the square root of the determinant of the Fisher information matrix. From (7), we obtain the Fisher information matrix of θ0,θ1,θ2 as(16)I=Iθ0,θ1,θ2=n0θ02000n1θ12000n2θ22.From Theorem1, we have ni=n⋅θi/θ0+θ1+θ2,i=0,1,2, so Iθ0,θ1,θ2 can be written as(17)I=n1θ0θ0001θ1θ0001θ2θ.Thus, the Jeffreys prior is given by(18)π2θ0,θ1,θ2∝1θ0θ1θ2θ3.Theorem 2.
Based on the Jeffreys priorπ2θ0,θ1,θ2, the joint posterior distribution of θ0,θ1,θ2 is proper.Proof.
From (6) and (7), we obtain the joint posterior distribution of θ0,θ1,θ2 based on π2θ0,θ1,θ2 as(19)π2θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2dθ0dθ1dθ2,∝θ−3/2θ0n0−1/2θ1n1−1/2θ2n2−1/2e−Aθ/λ.
Integratingπ2θ0,θ1,θ2|x with respect to θ0,θ1,andθ2, we obtain(20)∫0∞∫0∞∫0∞θ−3/2θ0n0−1/2θ1n1−1/2θ2n2−1/2e−Aθ/λdθ0dθ1dθ2,=∫0∞∫0∞∫0∞θ0θn0−1/2θ1θn1−1/2θ2θn2−1/2θn−3exp−Aλθdθ0dθ1dθ2,=∫0<θ0/θ+θ1/θ<1∏i=01θiθni−1/21−∑i=01θiθn2−1/2dθ0θdθ1θ⋅∫0∞θn−1exp−Aλθdθ,=Bn0+12,n1+n2+1Bn1+12,n2+12ΓnA/λn<∞,where A=∑i=1neλxi−1 and B⋅,⋅ is a beta function.
Thus, the joint posterior distribution ofθ0,θ1,θ2 based on π2θ0,θ1,θ2 is proper.


### 4.3. Reference Priors

Bernardo [18] and Berger and Bernardo [19] proposed the reference prior which plays a vital role in the objective Bayesian inference. We set μ0≡θ=θ0+θ1+θ2, μ1=θ0/θ, and μ2=θ1/θ; the transformation from θ0,θ1,θ2 to μ0,μ1,μ2 is one-to-one with the inverse transformation θ0=μ0μ1, θ1=μ0μ2, and θ2=μ01−μ1−μ2. The Jacobian matrix of the transformation has the form(21)J=μ1μ00μ20μ01−μ1−μ2−μ0−μ0,0<μ0<∞,0<μ1+μ2<1.The likelihood function (3) becomes(22)Lx;λ,μ0,μ1,μ2=μ1n0μ2n11−μ1−μ2n2μ0nexpλ∑lxl−μ0λ∑leλxl−1.The Fisher information matrix ofμ0,μ1,μ2 can be written as(23)I1=J′IJ=n1/μ020001μ1+11−μ1−μ211−μ1−μ2011−μ1−μ21μ2+11−μ1−μ2.Theorem 3.
(i)
Under the ordering groupsμ0,μ1,μ2 and μ1,μ2,μ0, the reference priors are the same, which is given by ωR1μ0,μ1,μ2=1/μ02μ1μ21−μ1−μ2; the corresponding reference prior for θ0,θ1,θ2 is π2θ0,θ1,θ2=1/θ0θ1θ2θ3(ii)
Under the ordering groupsμ0,μ1,μ2, μ0,μ2,μ1, μ1,μ0,μ2, and μ1,μ2,μ0, the reference priors are the same, which is given by ωR2μ0,μ1,μ2=1/μ02μ1μ21−μ1−μ21−μ1; the corresponding reference prior for θ0,θ1,θ2 is π3θ0,θ1,θ2=1/θ2θ0θ1θ2θ1+θ2(iii)
Under the ordering groupsμ2,μ0,μ1 and μ2,μ1,μ0, the reference priors are the same, which is given by ωR3μ0,μ1,μ2=1/μ02μ1μ21−μ1−μ21−μ2; the corresponding reference prior for θ0,θ1,θ2 is π4θ0,θ1,θ2=1/θ2θ0θ1θ2θ0+θ2Proof.
(i)
The Fisher information matrix ofμ0,μ1,μ2 is(24)I1=∑1100∑22,where∑11=n/μ02 and ∑22=n1/μ1+1/1−μ1−μ21/1−μ1−μ21/1−μ1−μ21/μ2+1/1−μ1−μ2.The reference prior for the ordering groupsμ0,μ1,μ2 and μ1,μ2,μ0 is the same as in [21], which is given by(25)ωR1μ0,μ1,μ2∝∑111/2∑221/2∝1μ02μ1μ21−μ1−μ2.(ii)
The inverse ofI1 is(26)H=1nμ02000μ11−μ1−μ1μ20−μ1μ2μ21−μ2.(iii)
According the notations in [18], we obtain h1=1/μ02, h2=1/μ11−μ1, and h3=1−μ1/μ21−μ1−μ2.Choose the compact setsΩk=μ0,μ1,μ2|a0k<μ0<b0k,a1k<μ1,a2k<μ2,μ1+μ2<dk, such that a0k,a1k,a2k⟶0, b0k⟶∞, and dk⟶1, as k⟶∞. Then, we have(27)πkμ0,μ1,μ2=h1h2h3∫a0kb0kh1dμ0⋅∫a1kdk−μ20h2dμ1⋅∫a2kdk−μ1h3dμ2IΩkμ0,μ1,μ2,where∫a0kb0kh1dμ0=∫a0kb0k1/μ0dμ0=logb0k−loga0k:(28)∫a1kdk−μ20h2dμ1=∫a1kdk−μ201μ11−μ1dμ1=−arcsin1−2dk−μ20+arcsin1−2a1k,∫a2kdk−μ1h3dμ2=∫a2kdk−μ11−μ1μ21−μ1−μ2dμ2,=1−μ11/2−arcsin1−μ1−2dk−μ11−μ1+arcsin1−μ1−2a2k1−μ1.Then, we get the reference prior as(29)ωR2μ0,μ1,μ2=limk⟶∞πkμ0,μ1,μ2πkμ0∗,μ1∗,μ2∗∝1μ02μ1μ21−μ1−μ21−μ1,whereμ0∗,μ1∗,μ2∗ is an inner point of Ωk.Similarly, under the ordering groupμ0,μ2,μ1, the reference prior is ωR2μ0,μ1,μ2.The Fisher information matrix ofμ1,μ0,μ2 is(30)I2=n1μ1+11−μ1−μ2011−μ1−μ201μ02011−μ1−μ201μ2+11−μ1−μ2.The inverse ofI2 is(31)H1=1nμ11−μ10−μ1μ20μ020−μ1μ20μ21−μ2.Similarly, we obtainh1=1/μ11−μ1, h2=1/μ02, and h3=1−μ1/μ21−μ1−μ2.Choose the compact setsΩk=μ1,μ0,μ2|a0k<μ1,a1k<μ0<b1k,a2k<μ2,μ1+μ2<dk, such that a0k,a1k,a2k⟶0, b1k⟶∞, and dk⟶1, as k⟶∞. Then, we have(32)πkμ1,μ0,μ2=h1h2h3∫a0kdk−u20h1dμ1⋅∫a1kb1kh2dμ0⋅∫a2kdk−μ1h3dμ2IΩkμ1,μ0,μ2,where∫a0kdk−u20h1dμ1=∫a0kdk−u201/μ11−μ1dμ1=−arcsin1−2dk−u20+arcsin1−2a0k,(33)∫a1kb1kh2dμ0=∫a1kb1k1μ0dμ0=logb1k−loga1k,∫a2kdk−μ1h3dμ2=∫a2kdk−μ11−μ1μ21−μ1−μ2dμ2,=1−μ11/2−arcsin1−μ1−2dk−u11−μ1+arcsin1−μ1−2a2k1−μ1.Letμ1∗,μ0∗,μ2∗ be an inner point of Ωk; we get the reference prior as(34)ωR2μ0,μ1,μ2=limk⟶∞πkμ1,μ0,μ2πkμ1∗,μ0∗,μ2∗∝1μ02μ1μ21−μ1−μ21−μ1.Similarly, under the ordering groupμ1,μ2,μ0, the reference prior is ωR2μ0,μ1,μ2.(v)
The Fisher information matrix ofμ2,μ1,μ0 is(35)I3=n1μ2+11−μ1−μ211−μ1−μ2011−μ1−μ21μ1+11−μ1−μ20001μ02.
The inverse ofI3 is(36)H2=1nμ21−μ2−μ1μ20−μ1μ2μ11−μ1000μ02.
Then, we obtainh1=1/μ21−μ2, h2=1−μ2/μ11−μ1−μ2, and h3=1/μ02.
Choose the compact setsΩk=μ2,μ1,μ0|a0k<μ2,a1k<μ1,μ2+μ1<dk,a2k<μ0<b2k, such that a0k,a1k,a2k⟶0, b2k⟶∞, and dk⟶1, as k⟶∞. Then, we have(37)πkμ2,μ1,μ0=h1h2h3∫a0kdk−u10h1dμ2⋅∫a1kdk−μ2h2dμ1⋅∫a2kb2kh3dμ0IΩkμ2,μ1,μ0,where ∫a0kdk−u10h1dμ2=∫a0kdk−u101/μ21−μ2dμ2=−arcsin1−2dk−u10+arcsin1−2a0k,(38)∫a1kdk−μ2h2dμ1=∫a1kdk−μ21−μ2μ11−μ1−μ2dμ1,=1−μ21/2−arcsin1−μ2−2dk−u21−μ2+arcsin1−μ2−2a1k1−μ2,∫a2kb2kh3dμ0=∫a2kb2k1μ0dμ0=logb2k−loga2k.
Letμ2∗,μ1∗,μ0∗ be an inner point of Ωk, we obtain the reference prior as(39)ωR3μ0,μ1,μ2=limk⟶∞πkμ2,μ1,μ0πkμ2∗,μ1∗,μ0∗∝1μ02μ1μ21−μ1−μ21−μ2.
Similarly, under the ordering groupμ2,μ0,μ1, the reference prior is ωR3μ0,μ1,μ2. According to the one-to-one transformation from μ0,μ1,μ2 to θ0,θ1,θ2, we can obtain the reference priors π2μ0,μ1,μ2,π3μ0,μ1,μ2,π4μ0,μ1,μ2 from ωR1,ωR2,andωR3, respectively.Theorem 4.
Based on the reference priorsπ3θ0,θ1,θ2 and π4θ0,θ1,θ2, the posterior distributions of θ0,θ1,θ2 are proper.Proof.
The joint posterior distributions ofθ0,θ1,θ2 based on reference prior π3θ0,θ1,θ2 and π4θ0,θ1,θ2 are, respectively, as(40)π3θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2dθ0dθ1dθ2,∝θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ1+θ2−1/2exp−Aθλ,π4θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2dθ0dθ1dθ2,∝θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ0+θ2−1/2exp−Aθλ.
Integratingπ3θ0,θ1,θ2|x and π4θ0,θ1,θ2|x with respect to θ0,θ1,andθ2, respectively, we obtain(41)∫0∞∫0∞∫0∞θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ1+θ2−1/2exp−Aθλdθ0dθ1dθ2,=Bn0+12,n1+n2+12Bn1+12,n2+12ΓnA/λn<∞,∫0∞∫0∞∫0∞θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ0+θ2−1/2exp−Aθλdθ0dθ1dθ2,=Bn1+12,n0+n2+12Bn0+12,n2+12ΓnA/λn<∞.
Thus, the posterior distributions ofθ0,θ1,θ2 based on π3θ0,θ1,θ2 and π4θ0,θ1,θ2 are proper.


### 4.4. Bayesian Estimates

The joint posterior distributions ofθ0,θ1,θ2 based on π1,π2,π3,andπ4 are, respectively, as(42)π1θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π1θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π1θ0,θ1,θ2dθ0dθ1dθ2,where(43)∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π1θ0,θ1,θ2dθ0dθ1dθ2=w1w2expλ∑lxl∫0<θ0/θ+θ1/θ<1∏i=01θiθni+ci−11−∑i=01θiθn2+c2−1dθ0θdθ1θ,∫0∞θn+a+1e−A/λ+bθdθ=w1w2expλ∑lxlBn0+c0,n1+c1+n2+c2Bn1+c1,n2+c2Γn+a+2A/λ+bn+a+2,where w1=Γ∑i=02ciba−c0−c1−c2/Γa and w2=∏i=02bci/Γci.Thus, we obtain(44)π1θ0,θ1,θ2|x=θa−c0−c1−c2θ0n0+c0−1θ1n1+c1−1θ2n2+c2−1exp−A/λ+bθBn0+c0,n1+c1+n2+c2Bn1+c1,n2+c2Γn+a/A/λ+bn+a.Similarly,(45)π2θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2dθ0dθ1dθ2,where(46)∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2dθ0dθ1dθ2,=∫0∞∫0∞∫0∞θ−3/2θ0n0−1/2θ1n1−1/2θ2n2−1/2expλ∑lxl−Aλθdθ0dθ1dθ2,=expλ∑lxlBn0+12,n1+n2+1Bn1+12,n2+12ΓnA/λn.We obtain(47)π2θ0,θ1,θ2|x=θ−3/2θ0n0−1/2θ1n1−1/2θ2n2−1/2exp−Aθ/λBn0+1/2,n1+n2+1Bn1+1/2,n2+1/2Γn/A/λn,π3θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2dθ0dθ1dθ2,where(48)∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2dθ0dθ1dθ2,=∫0∞∫0∞∫0∞θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ1+θ2−1/2expλ∑lxl−Aθλdθ0dθ1dθ2,=expλ∑lxlBn0+12,n1+n2+12Bn1+12,n2+12ΓnA/λn.We obtain(49)π3θ0,θ1,θ2|x=θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ1+θ2−1/2exp−Aθ/λBn0+1/2,n1+n2+1/2Bn1+1/2,n2+1/2Γn/A/λn,π4θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2dθ0dθ1dθ2,where(50)∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2dθ0dθ1dθ2,=∫0∞∫0∞∫0∞θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ0+θ2−1/2expλ∑lxl−Aθλdθ0dθ1dθ2,=expλ∑lxlBn1+12,n0+n2+12Bn0+12,n2+12ΓnA/λn.Then, we have(51)π4θ0,θ1,θ2|x=θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ0+θ2−1/2exp−Aθ/λBn1+1/2,n0+n2+1/2Bn0+1/2,n2+1/2Γn/A/λn.From (9)–(12), we get the Bayesian estimates of parameters θ0,θ1,θ2,andθ against squared error loss function based on π1,π2,π3,andπ4, respectively, which are listed in Table 1.Table 1
Bayesian estimates of parameters based on different priors.
Priorθ0θ1θ2θπ1λn0+c0n+a/n+c0+c1+c2A+bλλn1+c1n+a/n+c0+c1+c2A+bλλn2+c2n+a/n+c0+c1+c2bλ+An+aλ/A+bλπ2nλ2n0+1/A2n+3nλ2n1+1/A2n+3nλ2n2+1/A2n+3nλ/Aπ3nλ2n0+1/2An+1nλ2n1+2n2+12n1+1/4An+1n1+n2+1nλ2n1+2n2+12n2+1/4An+1n1+n2+1nλ/Aπ4nλ2n0+2n2+12n0+1/4An+1n0+n2+1nλ2n1+1/2An+1nλ2n0+2n2+12n2+1/2An+1n0+n2+1nλ/A


### 4.5. HPD Credible Intervals

The HPD credible intervals of parametersθ0,θ1,θ2,andθ can be constructed by the Monte Carlo method studied by Chen and Shao [22].Step 1: given the value ofn and the observed data x1,x2,⋯,xn, compute the Bayesian estimates of θ^0,θ^1,θ^2,andθ^ based on π1,π2,π3,andπ4, respectively.Step 2: repeat Step 1M times; we obtain M sets of the values θ^0,θ^1,θ^2,andθ^ based on π1,π2,π3,andπ4, respectively. Arrange them in the ascending order, we obtain(52)θ^0πk1<⋯<θ^0πkM,θ^1πk1<⋯<θ^1πkM,θ^2πk1<⋯<θ^2πkM,θ^πk1<⋯<θ^πkM,k=1,2,3,4.Step 3: compute the CIs at1−γ confidence level as(53)θ^vπkw,θ^vπkw+1−γM,θ^πkw,θ^πkw+1−γM,v=0,1,2;w=1,2,⋯,M−1−γM;k=1,2,3,4.Step 4
: the HPD CIs forθv,v=0,1,2, and θ are the shortest intervals among θ^vπkw,θ^vπkw+1−γM,θ^πkw,θ^πkw+1−γM, and w=1,2,⋯,M−1−γM, respectively.


## 4.1. Conjugate Prior

In this section, we suppose the shape parameterλ is known. Denote θ=θ0+θ1+θ2, which has a Gamma prior with hyperparameters a and b as(13)πθ=baΓaθa−1e−bθ,a>0,b>0,θ>0.Due toθ0/θ+θ1/θ+θ2/θ=1, so given θ, θ1/θ,θ2/θ follows a Dirichlet prior with hyper parameters c0,c1, and c2, that is,(14)πDθ1θ,θ2θ|θ=Γ∑i=02ci∏i=02Γci∏i=02θiθci−1,θi>0,ci>0,i=0,1,2.Therefore, the joint prior ofθ0,θ1,andθ2 becomes(15)π1θ0,θ1,θ2;a,b,c0,c1,c2=ΓcΓabθa−c∏i=02bciΓciθici−1exp−bθi,where c=c0+c1+c2.


## 4.2. Jeffreys Prior

According to Jeffreys [20], Jeffreys prior is proportional to the square root of the determinant of the Fisher information matrix. From (7), we obtain the Fisher information matrix of θ0,θ1,θ2 as(16)I=Iθ0,θ1,θ2=n0θ02000n1θ12000n2θ22.From Theorem1, we have ni=n⋅θi/θ0+θ1+θ2,i=0,1,2, so Iθ0,θ1,θ2 can be written as(17)I=n1θ0θ0001θ1θ0001θ2θ.Thus, the Jeffreys prior is given by(18)π2θ0,θ1,θ2∝1θ0θ1θ2θ3.Theorem 2.
Based on the Jeffreys priorπ2θ0,θ1,θ2, the joint posterior distribution of θ0,θ1,θ2 is proper.Proof.
From (6) and (7), we obtain the joint posterior distribution of θ0,θ1,θ2 based on π2θ0,θ1,θ2 as(19)π2θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2dθ0dθ1dθ2,∝θ−3/2θ0n0−1/2θ1n1−1/2θ2n2−1/2e−Aθ/λ.
Integratingπ2θ0,θ1,θ2|x with respect to θ0,θ1,andθ2, we obtain(20)∫0∞∫0∞∫0∞θ−3/2θ0n0−1/2θ1n1−1/2θ2n2−1/2e−Aθ/λdθ0dθ1dθ2,=∫0∞∫0∞∫0∞θ0θn0−1/2θ1θn1−1/2θ2θn2−1/2θn−3exp−Aλθdθ0dθ1dθ2,=∫0<θ0/θ+θ1/θ<1∏i=01θiθni−1/21−∑i=01θiθn2−1/2dθ0θdθ1θ⋅∫0∞θn−1exp−Aλθdθ,=Bn0+12,n1+n2+1Bn1+12,n2+12ΓnA/λn<∞,where A=∑i=1neλxi−1 and B⋅,⋅ is a beta function.
Thus, the joint posterior distribution ofθ0,θ1,θ2 based on π2θ0,θ1,θ2 is proper.


## 4.3. Reference Priors

Bernardo [18] and Berger and Bernardo [19] proposed the reference prior which plays a vital role in the objective Bayesian inference. We set μ0≡θ=θ0+θ1+θ2, μ1=θ0/θ, and μ2=θ1/θ; the transformation from θ0,θ1,θ2 to μ0,μ1,μ2 is one-to-one with the inverse transformation θ0=μ0μ1, θ1=μ0μ2, and θ2=μ01−μ1−μ2. The Jacobian matrix of the transformation has the form(21)J=μ1μ00μ20μ01−μ1−μ2−μ0−μ0,0<μ0<∞,0<μ1+μ2<1.The likelihood function (3) becomes(22)Lx;λ,μ0,μ1,μ2=μ1n0μ2n11−μ1−μ2n2μ0nexpλ∑lxl−μ0λ∑leλxl−1.The Fisher information matrix ofμ0,μ1,μ2 can be written as(23)I1=J′IJ=n1/μ020001μ1+11−μ1−μ211−μ1−μ2011−μ1−μ21μ2+11−μ1−μ2.Theorem 3.
(i)
Under the ordering groupsμ0,μ1,μ2 and μ1,μ2,μ0, the reference priors are the same, which is given by ωR1μ0,μ1,μ2=1/μ02μ1μ21−μ1−μ2; the corresponding reference prior for θ0,θ1,θ2 is π2θ0,θ1,θ2=1/θ0θ1θ2θ3(ii)
Under the ordering groupsμ0,μ1,μ2, μ0,μ2,μ1, μ1,μ0,μ2, and μ1,μ2,μ0, the reference priors are the same, which is given by ωR2μ0,μ1,μ2=1/μ02μ1μ21−μ1−μ21−μ1; the corresponding reference prior for θ0,θ1,θ2 is π3θ0,θ1,θ2=1/θ2θ0θ1θ2θ1+θ2(iii)
Under the ordering groupsμ2,μ0,μ1 and μ2,μ1,μ0, the reference priors are the same, which is given by ωR3μ0,μ1,μ2=1/μ02μ1μ21−μ1−μ21−μ2; the corresponding reference prior for θ0,θ1,θ2 is π4θ0,θ1,θ2=1/θ2θ0θ1θ2θ0+θ2Proof.
(i)
The Fisher information matrix ofμ0,μ1,μ2 is(24)I1=∑1100∑22,where∑11=n/μ02 and ∑22=n1/μ1+1/1−μ1−μ21/1−μ1−μ21/1−μ1−μ21/μ2+1/1−μ1−μ2.The reference prior for the ordering groupsμ0,μ1,μ2 and μ1,μ2,μ0 is the same as in [21], which is given by(25)ωR1μ0,μ1,μ2∝∑111/2∑221/2∝1μ02μ1μ21−μ1−μ2.(ii)
The inverse ofI1 is(26)H=1nμ02000μ11−μ1−μ1μ20−μ1μ2μ21−μ2.(iii)
According the notations in [18], we obtain h1=1/μ02, h2=1/μ11−μ1, and h3=1−μ1/μ21−μ1−μ2.Choose the compact setsΩk=μ0,μ1,μ2|a0k<μ0<b0k,a1k<μ1,a2k<μ2,μ1+μ2<dk, such that a0k,a1k,a2k⟶0, b0k⟶∞, and dk⟶1, as k⟶∞. Then, we have(27)πkμ0,μ1,μ2=h1h2h3∫a0kb0kh1dμ0⋅∫a1kdk−μ20h2dμ1⋅∫a2kdk−μ1h3dμ2IΩkμ0,μ1,μ2,where∫a0kb0kh1dμ0=∫a0kb0k1/μ0dμ0=logb0k−loga0k:(28)∫a1kdk−μ20h2dμ1=∫a1kdk−μ201μ11−μ1dμ1=−arcsin1−2dk−μ20+arcsin1−2a1k,∫a2kdk−μ1h3dμ2=∫a2kdk−μ11−μ1μ21−μ1−μ2dμ2,=1−μ11/2−arcsin1−μ1−2dk−μ11−μ1+arcsin1−μ1−2a2k1−μ1.Then, we get the reference prior as(29)ωR2μ0,μ1,μ2=limk⟶∞πkμ0,μ1,μ2πkμ0∗,μ1∗,μ2∗∝1μ02μ1μ21−μ1−μ21−μ1,whereμ0∗,μ1∗,μ2∗ is an inner point of Ωk.Similarly, under the ordering groupμ0,μ2,μ1, the reference prior is ωR2μ0,μ1,μ2.The Fisher information matrix ofμ1,μ0,μ2 is(30)I2=n1μ1+11−μ1−μ2011−μ1−μ201μ02011−μ1−μ201μ2+11−μ1−μ2.The inverse ofI2 is(31)H1=1nμ11−μ10−μ1μ20μ020−μ1μ20μ21−μ2.Similarly, we obtainh1=1/μ11−μ1, h2=1/μ02, and h3=1−μ1/μ21−μ1−μ2.Choose the compact setsΩk=μ1,μ0,μ2|a0k<μ1,a1k<μ0<b1k,a2k<μ2,μ1+μ2<dk, such that a0k,a1k,a2k⟶0, b1k⟶∞, and dk⟶1, as k⟶∞. Then, we have(32)πkμ1,μ0,μ2=h1h2h3∫a0kdk−u20h1dμ1⋅∫a1kb1kh2dμ0⋅∫a2kdk−μ1h3dμ2IΩkμ1,μ0,μ2,where∫a0kdk−u20h1dμ1=∫a0kdk−u201/μ11−μ1dμ1=−arcsin1−2dk−u20+arcsin1−2a0k,(33)∫a1kb1kh2dμ0=∫a1kb1k1μ0dμ0=logb1k−loga1k,∫a2kdk−μ1h3dμ2=∫a2kdk−μ11−μ1μ21−μ1−μ2dμ2,=1−μ11/2−arcsin1−μ1−2dk−u11−μ1+arcsin1−μ1−2a2k1−μ1.Letμ1∗,μ0∗,μ2∗ be an inner point of Ωk; we get the reference prior as(34)ωR2μ0,μ1,μ2=limk⟶∞πkμ1,μ0,μ2πkμ1∗,μ0∗,μ2∗∝1μ02μ1μ21−μ1−μ21−μ1.Similarly, under the ordering groupμ1,μ2,μ0, the reference prior is ωR2μ0,μ1,μ2.(v)
The Fisher information matrix ofμ2,μ1,μ0 is(35)I3=n1μ2+11−μ1−μ211−μ1−μ2011−μ1−μ21μ1+11−μ1−μ20001μ02.
The inverse ofI3 is(36)H2=1nμ21−μ2−μ1μ20−μ1μ2μ11−μ1000μ02.
Then, we obtainh1=1/μ21−μ2, h2=1−μ2/μ11−μ1−μ2, and h3=1/μ02.
Choose the compact setsΩk=μ2,μ1,μ0|a0k<μ2,a1k<μ1,μ2+μ1<dk,a2k<μ0<b2k, such that a0k,a1k,a2k⟶0, b2k⟶∞, and dk⟶1, as k⟶∞. Then, we have(37)πkμ2,μ1,μ0=h1h2h3∫a0kdk−u10h1dμ2⋅∫a1kdk−μ2h2dμ1⋅∫a2kb2kh3dμ0IΩkμ2,μ1,μ0,where ∫a0kdk−u10h1dμ2=∫a0kdk−u101/μ21−μ2dμ2=−arcsin1−2dk−u10+arcsin1−2a0k,(38)∫a1kdk−μ2h2dμ1=∫a1kdk−μ21−μ2μ11−μ1−μ2dμ1,=1−μ21/2−arcsin1−μ2−2dk−u21−μ2+arcsin1−μ2−2a1k1−μ2,∫a2kb2kh3dμ0=∫a2kb2k1μ0dμ0=logb2k−loga2k.
Letμ2∗,μ1∗,μ0∗ be an inner point of Ωk, we obtain the reference prior as(39)ωR3μ0,μ1,μ2=limk⟶∞πkμ2,μ1,μ0πkμ2∗,μ1∗,μ0∗∝1μ02μ1μ21−μ1−μ21−μ2.
Similarly, under the ordering groupμ2,μ0,μ1, the reference prior is ωR3μ0,μ1,μ2. According to the one-to-one transformation from μ0,μ1,μ2 to θ0,θ1,θ2, we can obtain the reference priors π2μ0,μ1,μ2,π3μ0,μ1,μ2,π4μ0,μ1,μ2 from ωR1,ωR2,andωR3, respectively.Theorem 4.
Based on the reference priorsπ3θ0,θ1,θ2 and π4θ0,θ1,θ2, the posterior distributions of θ0,θ1,θ2 are proper.Proof.
The joint posterior distributions ofθ0,θ1,θ2 based on reference prior π3θ0,θ1,θ2 and π4θ0,θ1,θ2 are, respectively, as(40)π3θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2dθ0dθ1dθ2,∝θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ1+θ2−1/2exp−Aθλ,π4θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2dθ0dθ1dθ2,∝θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ0+θ2−1/2exp−Aθλ.
Integratingπ3θ0,θ1,θ2|x and π4θ0,θ1,θ2|x with respect to θ0,θ1,andθ2, respectively, we obtain(41)∫0∞∫0∞∫0∞θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ1+θ2−1/2exp−Aθλdθ0dθ1dθ2,=Bn0+12,n1+n2+12Bn1+12,n2+12ΓnA/λn<∞,∫0∞∫0∞∫0∞θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ0+θ2−1/2exp−Aθλdθ0dθ1dθ2,=Bn1+12,n0+n2+12Bn0+12,n2+12ΓnA/λn<∞.
Thus, the posterior distributions ofθ0,θ1,θ2 based on π3θ0,θ1,θ2 and π4θ0,θ1,θ2 are proper.


## 4.4. Bayesian Estimates

The joint posterior distributions ofθ0,θ1,θ2 based on π1,π2,π3,andπ4 are, respectively, as(42)π1θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π1θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π1θ0,θ1,θ2dθ0dθ1dθ2,where(43)∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π1θ0,θ1,θ2dθ0dθ1dθ2=w1w2expλ∑lxl∫0<θ0/θ+θ1/θ<1∏i=01θiθni+ci−11−∑i=01θiθn2+c2−1dθ0θdθ1θ,∫0∞θn+a+1e−A/λ+bθdθ=w1w2expλ∑lxlBn0+c0,n1+c1+n2+c2Bn1+c1,n2+c2Γn+a+2A/λ+bn+a+2,where w1=Γ∑i=02ciba−c0−c1−c2/Γa and w2=∏i=02bci/Γci.Thus, we obtain(44)π1θ0,θ1,θ2|x=θa−c0−c1−c2θ0n0+c0−1θ1n1+c1−1θ2n2+c2−1exp−A/λ+bθBn0+c0,n1+c1+n2+c2Bn1+c1,n2+c2Γn+a/A/λ+bn+a.Similarly,(45)π2θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2dθ0dθ1dθ2,where(46)∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π2θ0,θ1,θ2dθ0dθ1dθ2,=∫0∞∫0∞∫0∞θ−3/2θ0n0−1/2θ1n1−1/2θ2n2−1/2expλ∑lxl−Aλθdθ0dθ1dθ2,=expλ∑lxlBn0+12,n1+n2+1Bn1+12,n2+12ΓnA/λn.We obtain(47)π2θ0,θ1,θ2|x=θ−3/2θ0n0−1/2θ1n1−1/2θ2n2−1/2exp−Aθ/λBn0+1/2,n1+n2+1Bn1+1/2,n2+1/2Γn/A/λn,π3θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2dθ0dθ1dθ2,where(48)∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π3θ0,θ1,θ2dθ0dθ1dθ2,=∫0∞∫0∞∫0∞θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ1+θ2−1/2expλ∑lxl−Aθλdθ0dθ1dθ2,=expλ∑lxlBn0+12,n1+n2+12Bn1+12,n2+12ΓnA/λn.We obtain(49)π3θ0,θ1,θ2|x=θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ1+θ2−1/2exp−Aθ/λBn0+1/2,n1+n2+1/2Bn1+1/2,n2+1/2Γn/A/λn,π4θ0,θ1,θ2|x=Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2dθ0dθ1dθ2,where(50)∫0∞∫0∞∫0∞Lx;λ,θ0,θ1,θ2π4θ0,θ1,θ2dθ0dθ1dθ2,=∫0∞∫0∞∫0∞θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ0+θ2−1/2expλ∑lxl−Aθλdθ0dθ1dθ2,=expλ∑lxlBn1+12,n0+n2+12Bn0+12,n2+12ΓnA/λn.Then, we have(51)π4θ0,θ1,θ2|x=θ−1θ0n0−1/2θ1n1−1/2θ2n2−1/2θ0+θ2−1/2exp−Aθ/λBn1+1/2,n0+n2+1/2Bn0+1/2,n2+1/2Γn/A/λn.From (9)–(12), we get the Bayesian estimates of parameters θ0,θ1,θ2,andθ against squared error loss function based on π1,π2,π3,andπ4, respectively, which are listed in Table 1.Table 1
Bayesian estimates of parameters based on different priors.
Priorθ0θ1θ2θπ1λn0+c0n+a/n+c0+c1+c2A+bλλn1+c1n+a/n+c0+c1+c2A+bλλn2+c2n+a/n+c0+c1+c2bλ+An+aλ/A+bλπ2nλ2n0+1/A2n+3nλ2n1+1/A2n+3nλ2n2+1/A2n+3nλ/Aπ3nλ2n0+1/2An+1nλ2n1+2n2+12n1+1/4An+1n1+n2+1nλ2n1+2n2+12n2+1/4An+1n1+n2+1nλ/Aπ4nλ2n0+2n2+12n0+1/4An+1n0+n2+1nλ2n1+1/2An+1nλ2n0+2n2+12n2+1/2An+1n0+n2+1nλ/A


## 4.5. HPD Credible Intervals

The HPD credible intervals of parametersθ0,θ1,θ2,andθ can be constructed by the Monte Carlo method studied by Chen and Shao [22].Step 1: given the value ofn and the observed data x1,x2,⋯,xn, compute the Bayesian estimates of θ^0,θ^1,θ^2,andθ^ based on π1,π2,π3,andπ4, respectively.Step 2: repeat Step 1M times; we obtain M sets of the values θ^0,θ^1,θ^2,andθ^ based on π1,π2,π3,andπ4, respectively. Arrange them in the ascending order, we obtain(52)θ^0πk1<⋯<θ^0πkM,θ^1πk1<⋯<θ^1πkM,θ^2πk1<⋯<θ^2πkM,θ^πk1<⋯<θ^πkM,k=1,2,3,4.Step 3: compute the CIs at1−γ confidence level as(53)θ^vπkw,θ^vπkw+1−γM,θ^πkw,θ^πkw+1−γM,v=0,1,2;w=1,2,⋯,M−1−γM;k=1,2,3,4.Step 4
: the HPD CIs forθv,v=0,1,2, and θ are the shortest intervals among θ^vπkw,θ^vπkw+1−γM,θ^πkw,θ^πkw+1−γM, and w=1,2,⋯,M−1−γM, respectively.


## 5. Numerical Simulation and Illustrative Example

### 5.1. Simulation

Suppose the common shape parameterλ is known. The initial values for parameters λ,θ0,θ1,θ2 are 3,1,2,1. The initial values for the hyperparameters a,b,c0,c1,andc2 are all 0.001. Take the sample size n = 10, 20, 30, and 50. Generate the random samples x1,x2,⋯,xn from MOGPλ,θ0,θ1,θ2 by the following steps:Step 1: for a fixed valuen, generate n samples u01,u02,⋯,u0n from GPλ,θ0, u11,u12,⋯,u1n from GPλ,θ1, and u21,u22,⋯,u2n from GPλ,θ2. Then, we obtain T1l=minu0l,u1l and T2l=minu0l,u2l,l=1,2,⋯,n.Step 2: computexl,δ0l,δ1l,δ2l,l=1,2,⋯,n, where xl=minT1l,T2l, δ0l=IT1l=T2l, δ1l=IT1l<T2l, and δ2l=IT1l>T2l.Repeat the procedures 10,000 times; we get the values of the mean squared errors (MSEs) of the MLEs, the average lengths (ALs), and coverage probabilities (CPs) of the 95% Boot-P CIs, and the MSEs of the Bayesian estimates, the ALs, and CPs of the 95% HPD CIs, which are shown in Table2–5. From the results in Table 2–5, we can make the following conclusions.Table 2
MSEs, ALs, and CPs ofθ0,θ1,θ2,andθ (n = 10).
MethodPara.θ0θ1θ2θMLEMSE0.48580.80300.48650.9374Boot-AL2.24142.71462.22661.7920Boot-CP0.93390.92940.94050.9321Bayesπ1MSE0.48500.80120.48570.9340HPD-AL2.03882.51192.04251.9018HPD-CP0.96630.94400.96450.9369π2MSE0.40550.59030.40610.9374HPD-AL1.79802.21831.80161.9034HPD-CP0.95520.93990.95390.9335π3MSE0.46780.57320.39090.9374HPD-AL1.87972.21931.78501.9034HPD-CP0.94810.94050.95690.9460π4MSE0.37480.70420.37540.9374HPD-AL1.77242.31921.77601.9034HPD-CP0.95270.94680.95150.9405Table 3
MSEs, ALs, and CPs ofθ0,θ1,θ2,andθ (n = 20).
MethodPara.θ0θ1θ2θMLEMSE0.25050.45190.25230.6907Boot-AL1.57951.90481.58071.2957Boot-CP0.94880.94830.94120.9407Bayesπ1MSE0.25030.45120.25200.6893HPD-AL1.44341.75731.43821.3635HPD-CP0.98320.96920.98310.9415π2MSE0.23350.37660.23500.6907HPD-AL1.35121.64761.34621.3640HPD-CP0.97460.94470.97620.9409π3MSE0.25510.36620.22930.6907HPD-AL1.38341.64861.33981.3640HPD-CP0.96680.95060.97770.9598π4MSE0.22010.42600.22160.6907HPD-AL1.33991.68681.33481.3640HPD-CP0.96140.95250.96130.9498Table 4
MSEs, ALs, and CPs ofθ0,θ1,θ2,andθ (n = 30).
MethodPara.θ0θ1θ2θMLEMSE0.17520.33450.17710.6049Boot-AL1.28491.54511.28961.0510Boot-CP0.96510.95160.96540.9415Bayesπ1MSE0.17510.33410.17700.6040HPD-AL1.17101.43541.17271.1164HPD-CP0.99190.96290.99010.9427π2MSE0.16940.29220.17120.6049HPD-AL1.11971.37451.12131.1167HPD-CP0.98390.97230.98350.9418π3MSE0.18140.28490.16790.6049HPD-AL1.13771.37501.11771.1167HPD-CP0.97830.97700.98510.9615π4MSE0.16120.32280.16290.6049HPD-AL1.11321.39661.11481.1167HPD-CP0.98960.95810.98850.9638Table 5
MSEs, ALs, and CPs ofθ0,θ1,θ2,andθ (n = 50).
MethodPara.θ0θ1θ2θMLEMSE0.11580.24600.11610.5380Boot-AL0.99471.19811.00180.8227Boot-CP0.98290.95780.98310.9554Bayesπ1MSE0.11570.24580.11610.5375HPD-AL0.91181.10750.90710.8677HPD-CP0.99550.98220.99540.9724π2MSE0.11500.22430.11540.5380HPD-AL0.88741.07860.88280.8679HPD-CP0.98840.99000.98700.9702π3MSE0.12090.21960.11370.5380HPD-AL0.89611.07910.88110.8679HPD-CP0.98130.99330.99010.9721π4MSE0.11050.24170.11090.5380HPD-AL0.88421.08920.87960.8679HPD-CP0.99380.97890.99310.9717The MSEs of MLEs and Bayesian estimates decrease as the sample size increases. For given sample sizen, the Bayesian estimates based on π1,π2,andπ4 are smaller than the MSEs of MLEs. The MSEs of Bayesian estimates of θ0andθ2 based on π4 are smaller than that based on π1,π2,andπ3. The MSEs of Bayesian estimates of θ1 based on π3 are smaller than that based on π1,π2,andπ4. The MSEs of Bayesian estimates of θ based on π1 are smaller than that based on π2,π3,andπ4.The CPs of Boot-P and HPD CIs are all close to 0.95. The ALs of Boot-P and HPD CIs decrease; the associated CPs increase when the sample size increases. The CPs of HPD CIs based on Bayesian estimates are larger than the CPs of Boot-P CIs based on MLEs.


### 5.2. Illustrative Analysis

#### 5.2.1. Simulated Data

For illustrative purposes, with initial value for parametersλ,θ0,θ1,θ2 as 3,1,2,1, we use the procedures mentioned above to generate U0,U1,andU2 from GP3,1, GP3,2, and GP3,1, respectively. We then get T1=minU0,U1 and T2=minU0,U2; the latent lifetime of the system is minT1,T2. The simulated data are listed in Table 6. The MLEs, Bayesian estimates, and associated 95% CIs for parameters θ0,θ1,θ2,andθ are shown in Table 7. From Table 7, all the MLEs and Bayesian estimates of θ0,θ1,θ2,θ are close to the true value.Table 6
The simulated data whenn = 25.
(0.00191)(0.00251)(0.00621)(0.03610)(0.06512)(0.06751)(0.11082)(0.14471)(0.15091)(0.16942)(0.17370)(0.18591)(0.19002)(0.22180)(0.23072)(0.25371)(0.25582)(0.27342)(0.27500)(0.33491)(0.35281)(0.37900)(0.38242)(0.38751)(0.53361)Table 7
Point estimates and 95% CIs ofθ0,θ1,θ2,andθ.
MethodPara.θ0θ1θ2θMLEMLE0.80291.92701.28474.0146Boot-CI(0.0408, 1.8099)(0.1891, 2.8803)(0.0642, 1.8178)(0.7671, 4.4112)Bayesπ1Bayes0.80291.92671.28454.0141HPD CI(0.0898, 1.6991)(0.3473, 2.9083)(0.0396, 1.7374)(0.8141, 4.1876)π2Bayes0.83321.89371.28774.0146HPD CI(0.0694, 1.7457)(0.3070, 2.8296)(0.0364, 1.7697)(0.7893, 4.4378)π3Bayes0.84921.88411.28124.0146HPD CI(0.0798, 1.7561)(0.1251, 2.5045)(0.0315, 1.4392)(0.6368, 4.1407)π4Bayes0.81891.93011.26564.0146HPD-CP(0.0638, 1.4011)(0.2448, 2.8554)(0.0456, 1.7418)(0.8474, 4.3484)


#### 5.2.2. Real Data

Use the procedures mentioned above to a real dataset. Kundu and Gupta [13] analyzed the football data of UEFA Champions’ League data which are presented in Table 1. From the data, T1 and T2 can be regarded as two dependent failure modes, and n0=7,n1=17,andn2=13. This data have been fitted by Marshall–Olkin bivariate Gompertz distribution (see Wang et al. [23]).The MLEs, Bayesian estimates, and associated 95% CIs for parametersθ0,θ1,θ2,andθ are shown in Table 8. From Tables 7 and 8, Bayesian estimates under different priors are close to MLEs, and the lengths of 95% Boot-p CIs associated to MLEs are longer than the lengths of 95% HPD CIs associated to Bayesian estimates.Table 8
Point estimates and 95% CIs ofθ0,θ1,θ2,andθ.
MethodPara.θ0θ1θ2θMLEMLE1.0882e-20.4664e-21.3214e-22.8760e-2Boot-CI(0.7354e-3, 1.2770e-2)(0.3244e-2, 2.4914e-2)(0.4077e-3, 1.5990e-2)(0.7324e-2, 3.5454e-2)Bayesπ1Bayes1.1882e-20.6879e-21.3758e-23.2520e-2HPD CI(0.2305e-2, 1.1210e-2)(0.3998e-2, 1.9825e-2)(0.1861e-2, 1.4366e-2)(1.0088e-2, 3.8182e-2)π2Bayes1.0832e-20.4856e-21.3073e-22.8760e-2HPD CI(0.8317e-3, 1.1676e-2)(0.2315e-2, 1.8702e-2)(0.8807e-3, 1.3700e-2)(0.5246e-2, 3.2002e-2)π3Bayes1.0974e-20.4817e-21.2969e-22.8760e-2HPD CI(0.8499e-3, 1.1535e-2)(0.3160e-2, 1.7152e-2)(0.8814e-3, 1.4144e-2)(0.6994e-2, 3.2140e-2)π4Bayes1.0803e-20.4919e-21.3038e-22.8760e-2HPD-CP(0.8949e-3, 1.1372e-2)(0.3365e-2, 1.8866e-2)(0.6968e-3, 1.4164e-2)(0.7224e-2, 3.0467e-2)


## 5.1. Simulation

Suppose the common shape parameterλ is known. The initial values for parameters λ,θ0,θ1,θ2 are 3,1,2,1. The initial values for the hyperparameters a,b,c0,c1,andc2 are all 0.001. Take the sample size n = 10, 20, 30, and 50. Generate the random samples x1,x2,⋯,xn from MOGPλ,θ0,θ1,θ2 by the following steps:Step 1: for a fixed valuen, generate n samples u01,u02,⋯,u0n from GPλ,θ0, u11,u12,⋯,u1n from GPλ,θ1, and u21,u22,⋯,u2n from GPλ,θ2. Then, we obtain T1l=minu0l,u1l and T2l=minu0l,u2l,l=1,2,⋯,n.Step 2: computexl,δ0l,δ1l,δ2l,l=1,2,⋯,n, where xl=minT1l,T2l, δ0l=IT1l=T2l, δ1l=IT1l<T2l, and δ2l=IT1l>T2l.Repeat the procedures 10,000 times; we get the values of the mean squared errors (MSEs) of the MLEs, the average lengths (ALs), and coverage probabilities (CPs) of the 95% Boot-P CIs, and the MSEs of the Bayesian estimates, the ALs, and CPs of the 95% HPD CIs, which are shown in Table2–5. From the results in Table 2–5, we can make the following conclusions.Table 2
MSEs, ALs, and CPs ofθ0,θ1,θ2,andθ (n = 10).
MethodPara.θ0θ1θ2θMLEMSE0.48580.80300.48650.9374Boot-AL2.24142.71462.22661.7920Boot-CP0.93390.92940.94050.9321Bayesπ1MSE0.48500.80120.48570.9340HPD-AL2.03882.51192.04251.9018HPD-CP0.96630.94400.96450.9369π2MSE0.40550.59030.40610.9374HPD-AL1.79802.21831.80161.9034HPD-CP0.95520.93990.95390.9335π3MSE0.46780.57320.39090.9374HPD-AL1.87972.21931.78501.9034HPD-CP0.94810.94050.95690.9460π4MSE0.37480.70420.37540.9374HPD-AL1.77242.31921.77601.9034HPD-CP0.95270.94680.95150.9405Table 3
MSEs, ALs, and CPs ofθ0,θ1,θ2,andθ (n = 20).
MethodPara.θ0θ1θ2θMLEMSE0.25050.45190.25230.6907Boot-AL1.57951.90481.58071.2957Boot-CP0.94880.94830.94120.9407Bayesπ1MSE0.25030.45120.25200.6893HPD-AL1.44341.75731.43821.3635HPD-CP0.98320.96920.98310.9415π2MSE0.23350.37660.23500.6907HPD-AL1.35121.64761.34621.3640HPD-CP0.97460.94470.97620.9409π3MSE0.25510.36620.22930.6907HPD-AL1.38341.64861.33981.3640HPD-CP0.96680.95060.97770.9598π4MSE0.22010.42600.22160.6907HPD-AL1.33991.68681.33481.3640HPD-CP0.96140.95250.96130.9498Table 4
MSEs, ALs, and CPs ofθ0,θ1,θ2,andθ (n = 30).
MethodPara.θ0θ1θ2θMLEMSE0.17520.33450.17710.6049Boot-AL1.28491.54511.28961.0510Boot-CP0.96510.95160.96540.9415Bayesπ1MSE0.17510.33410.17700.6040HPD-AL1.17101.43541.17271.1164HPD-CP0.99190.96290.99010.9427π2MSE0.16940.29220.17120.6049HPD-AL1.11971.37451.12131.1167HPD-CP0.98390.97230.98350.9418π3MSE0.18140.28490.16790.6049HPD-AL1.13771.37501.11771.1167HPD-CP0.97830.97700.98510.9615π4MSE0.16120.32280.16290.6049HPD-AL1.11321.39661.11481.1167HPD-CP0.98960.95810.98850.9638Table 5
MSEs, ALs, and CPs ofθ0,θ1,θ2,andθ (n = 50).
MethodPara.θ0θ1θ2θMLEMSE0.11580.24600.11610.5380Boot-AL0.99471.19811.00180.8227Boot-CP0.98290.95780.98310.9554Bayesπ1MSE0.11570.24580.11610.5375HPD-AL0.91181.10750.90710.8677HPD-CP0.99550.98220.99540.9724π2MSE0.11500.22430.11540.5380HPD-AL0.88741.07860.88280.8679HPD-CP0.98840.99000.98700.9702π3MSE0.12090.21960.11370.5380HPD-AL0.89611.07910.88110.8679HPD-CP0.98130.99330.99010.9721π4MSE0.11050.24170.11090.5380HPD-AL0.88421.08920.87960.8679HPD-CP0.99380.97890.99310.9717The MSEs of MLEs and Bayesian estimates decrease as the sample size increases. For given sample sizen, the Bayesian estimates based on π1,π2,andπ4 are smaller than the MSEs of MLEs. The MSEs of Bayesian estimates of θ0andθ2 based on π4 are smaller than that based on π1,π2,andπ3. The MSEs of Bayesian estimates of θ1 based on π3 are smaller than that based on π1,π2,andπ4. The MSEs of Bayesian estimates of θ based on π1 are smaller than that based on π2,π3,andπ4.The CPs of Boot-P and HPD CIs are all close to 0.95. The ALs of Boot-P and HPD CIs decrease; the associated CPs increase when the sample size increases. The CPs of HPD CIs based on Bayesian estimates are larger than the CPs of Boot-P CIs based on MLEs.


## 5.2. Illustrative Analysis

### 5.2.1. Simulated Data

For illustrative purposes, with initial value for parametersλ,θ0,θ1,θ2 as 3,1,2,1, we use the procedures mentioned above to generate U0,U1,andU2 from GP3,1, GP3,2, and GP3,1, respectively. We then get T1=minU0,U1 and T2=minU0,U2; the latent lifetime of the system is minT1,T2. The simulated data are listed in Table 6. The MLEs, Bayesian estimates, and associated 95% CIs for parameters θ0,θ1,θ2,andθ are shown in Table 7. From Table 7, all the MLEs and Bayesian estimates of θ0,θ1,θ2,θ are close to the true value.Table 6
The simulated data whenn = 25.
(0.00191)(0.00251)(0.00621)(0.03610)(0.06512)(0.06751)(0.11082)(0.14471)(0.15091)(0.16942)(0.17370)(0.18591)(0.19002)(0.22180)(0.23072)(0.25371)(0.25582)(0.27342)(0.27500)(0.33491)(0.35281)(0.37900)(0.38242)(0.38751)(0.53361)Table 7
Point estimates and 95% CIs ofθ0,θ1,θ2,andθ.
MethodPara.θ0θ1θ2θMLEMLE0.80291.92701.28474.0146Boot-CI(0.0408, 1.8099)(0.1891, 2.8803)(0.0642, 1.8178)(0.7671, 4.4112)Bayesπ1Bayes0.80291.92671.28454.0141HPD CI(0.0898, 1.6991)(0.3473, 2.9083)(0.0396, 1.7374)(0.8141, 4.1876)π2Bayes0.83321.89371.28774.0146HPD CI(0.0694, 1.7457)(0.3070, 2.8296)(0.0364, 1.7697)(0.7893, 4.4378)π3Bayes0.84921.88411.28124.0146HPD CI(0.0798, 1.7561)(0.1251, 2.5045)(0.0315, 1.4392)(0.6368, 4.1407)π4Bayes0.81891.93011.26564.0146HPD-CP(0.0638, 1.4011)(0.2448, 2.8554)(0.0456, 1.7418)(0.8474, 4.3484)


### 5.2.2. Real Data

Use the procedures mentioned above to a real dataset. Kundu and Gupta [13] analyzed the football data of UEFA Champions’ League data which are presented in Table 1. From the data, T1 and T2 can be regarded as two dependent failure modes, and n0=7,n1=17,andn2=13. This data have been fitted by Marshall–Olkin bivariate Gompertz distribution (see Wang et al. [23]).The MLEs, Bayesian estimates, and associated 95% CIs for parametersθ0,θ1,θ2,andθ are shown in Table 8. From Tables 7 and 8, Bayesian estimates under different priors are close to MLEs, and the lengths of 95% Boot-p CIs associated to MLEs are longer than the lengths of 95% HPD CIs associated to Bayesian estimates.Table 8
Point estimates and 95% CIs ofθ0,θ1,θ2,andθ.
MethodPara.θ0θ1θ2θMLEMLE1.0882e-20.4664e-21.3214e-22.8760e-2Boot-CI(0.7354e-3, 1.2770e-2)(0.3244e-2, 2.4914e-2)(0.4077e-3, 1.5990e-2)(0.7324e-2, 3.5454e-2)Bayesπ1Bayes1.1882e-20.6879e-21.3758e-23.2520e-2HPD CI(0.2305e-2, 1.1210e-2)(0.3998e-2, 1.9825e-2)(0.1861e-2, 1.4366e-2)(1.0088e-2, 3.8182e-2)π2Bayes1.0832e-20.4856e-21.3073e-22.8760e-2HPD CI(0.8317e-3, 1.1676e-2)(0.2315e-2, 1.8702e-2)(0.8807e-3, 1.3700e-2)(0.5246e-2, 3.2002e-2)π3Bayes1.0974e-20.4817e-21.2969e-22.8760e-2HPD CI(0.8499e-3, 1.1535e-2)(0.3160e-2, 1.7152e-2)(0.8814e-3, 1.4144e-2)(0.6994e-2, 3.2140e-2)π4Bayes1.0803e-20.4919e-21.3038e-22.8760e-2HPD-CP(0.8949e-3, 1.1372e-2)(0.3365e-2, 1.8866e-2)(0.6968e-3, 1.4164e-2)(0.7224e-2, 3.0467e-2)


## 5.2.1. Simulated Data

For illustrative purposes, with initial value for parametersλ,θ0,θ1,θ2 as 3,1,2,1, we use the procedures mentioned above to generate U0,U1,andU2 from GP3,1, GP3,2, and GP3,1, respectively. We then get T1=minU0,U1 and T2=minU0,U2; the latent lifetime of the system is minT1,T2. The simulated data are listed in Table 6. The MLEs, Bayesian estimates, and associated 95% CIs for parameters θ0,θ1,θ2,andθ are shown in Table 7. From Table 7, all the MLEs and Bayesian estimates of θ0,θ1,θ2,θ are close to the true value.Table 6
The simulated data whenn = 25.
(0.00191)(0.00251)(0.00621)(0.03610)(0.06512)(0.06751)(0.11082)(0.14471)(0.15091)(0.16942)(0.17370)(0.18591)(0.19002)(0.22180)(0.23072)(0.25371)(0.25582)(0.27342)(0.27500)(0.33491)(0.35281)(0.37900)(0.38242)(0.38751)(0.53361)Table 7
Point estimates and 95% CIs ofθ0,θ1,θ2,andθ.
MethodPara.θ0θ1θ2θMLEMLE0.80291.92701.28474.0146Boot-CI(0.0408, 1.8099)(0.1891, 2.8803)(0.0642, 1.8178)(0.7671, 4.4112)Bayesπ1Bayes0.80291.92671.28454.0141HPD CI(0.0898, 1.6991)(0.3473, 2.9083)(0.0396, 1.7374)(0.8141, 4.1876)π2Bayes0.83321.89371.28774.0146HPD CI(0.0694, 1.7457)(0.3070, 2.8296)(0.0364, 1.7697)(0.7893, 4.4378)π3Bayes0.84921.88411.28124.0146HPD CI(0.0798, 1.7561)(0.1251, 2.5045)(0.0315, 1.4392)(0.6368, 4.1407)π4Bayes0.81891.93011.26564.0146HPD-CP(0.0638, 1.4011)(0.2448, 2.8554)(0.0456, 1.7418)(0.8474, 4.3484)


## 5.2.2. Real Data

Use the procedures mentioned above to a real dataset. Kundu and Gupta [13] analyzed the football data of UEFA Champions’ League data which are presented in Table 1. From the data, T1 and T2 can be regarded as two dependent failure modes, and n0=7,n1=17,andn2=13. This data have been fitted by Marshall–Olkin bivariate Gompertz distribution (see Wang et al. [23]).The MLEs, Bayesian estimates, and associated 95% CIs for parametersθ0,θ1,θ2,andθ are shown in Table 8. From Tables 7 and 8, Bayesian estimates under different priors are close to MLEs, and the lengths of 95% Boot-p CIs associated to MLEs are longer than the lengths of 95% HPD CIs associated to Bayesian estimates.Table 8
Point estimates and 95% CIs ofθ0,θ1,θ2,andθ.
MethodPara.θ0θ1θ2θMLEMLE1.0882e-20.4664e-21.3214e-22.8760e-2Boot-CI(0.7354e-3, 1.2770e-2)(0.3244e-2, 2.4914e-2)(0.4077e-3, 1.5990e-2)(0.7324e-2, 3.5454e-2)Bayesπ1Bayes1.1882e-20.6879e-21.3758e-23.2520e-2HPD CI(0.2305e-2, 1.1210e-2)(0.3998e-2, 1.9825e-2)(0.1861e-2, 1.4366e-2)(1.0088e-2, 3.8182e-2)π2Bayes1.0832e-20.4856e-21.3073e-22.8760e-2HPD CI(0.8317e-3, 1.1676e-2)(0.2315e-2, 1.8702e-2)(0.8807e-3, 1.3700e-2)(0.5246e-2, 3.2002e-2)π3Bayes1.0974e-20.4817e-21.2969e-22.8760e-2HPD CI(0.8499e-3, 1.1535e-2)(0.3160e-2, 1.7152e-2)(0.8814e-3, 1.4144e-2)(0.6994e-2, 3.2140e-2)π4Bayes1.0803e-20.4919e-21.3038e-22.8760e-2HPD-CP(0.8949e-3, 1.1372e-2)(0.3365e-2, 1.8866e-2)(0.6968e-3, 1.4164e-2)(0.7224e-2, 3.0467e-2)


## 6. Conclusion

This paper discussed the point estimates and CIs for the parameters of the dependent competing risks’ model from MOGP distribution. We studied the appropriateness of the posteriors based on conjugate prior and Jeffreys and Reference priors, obtained the Bayesian estimates in closed forms, and constructed the associated HPD CIs. From the simulations results, the use of the Bayesian method can be recommended if the priors are available. The results of the illustrative analysis show that the proposed methods work well; from the lengths of CIs, we can conclude the Bayesian estimates are better than MLEs in general.



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*Source: 3988225-2022-05-28.xml*