# A Nonconstant Shape Parameter-Dependent Competing Risks’ Model in Accelerate Life Test Based on Adaptive Type-II Progressive Hybrid Censoring

Authors: Yan Wang; Yimin Shi; Min Wu
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

In this paper, the dependent competing risks’ model is considered in the constant-stress accelerated life test under the adaptive type-II progressive hybrid-censored scheme. The dependency between failure causes is modeled by Marshall–Olkin bivariate Gompertz distribution. The scale and shape parameters in the model both change with the stress levels, and the failure causes of some test units are unknown. Then, the maximum likelihood estimations and approximation confidence intervals of the unknown parameters are considered. And, the necessary and sufficient condition is established for the existence and uniqueness of the maximum likelihood estimations for unknown parameters. The Bayes approach is also employed to estimate the unknown parameters under suitable prior distributions. The Bayes estimations and highest posterior credible intervals of the unknown parameters are obtained. Finally, a simulation experiment has been performed to illustrate the methods proposed in this paper.


---

## Body

## 1. Introduction

The accelerated life test (ALT) has been proposed as an effective method to obtain failure information of the highly reliable units, which frequently appears in electronics, automation, or aerospace fields, and its test units are subjected to the higher-than-use stress levels. In the ALT, it is generally assumed that the scale parameters of life distribution change with the stress levels, while the other shape parameter is constant during the test. However, some research results have shown that the shape parameters change with the stress levels. Researchers such as Li et al. [1] discovered that the shape parameter of life distribution for electrolyte changes with the increase of voltage, when electrolyte is subjected to an electric test. Hiergeist et al. [2] discovered that the shape parameter of life distribution changes with the corresponding temperature in the capacitor test.In the competing risks’ model, there are more than one failure causes, and each cause possibly leads to the final failure of the unit. The unit operation may cause correlation among the failure causes, and failure causes may make the unit fail early. So, a failure cause is commonly associated with the other failure causes, and it is not usually possible to study the units with the isolated failure cause, and it is necessary to access each failure cause in the presence of other failure causes. To construct the dependence between failure causes, it is natural to consider a bivariate or multivariate distribution for the lifetimes. Among the multivariate distribution, the Marshall–Olkin family distributions in [3] are important bivariate distributions, and they have a parameter that describes the correlation structure between marginal distributions. Thus, they are used to describe the dependence between the failure causes. For example, Guan et al. [4] discussed the Bayes estimations for Marshall–Olkin bivariate exponential (MOBE) distribution. Shen and Xu [5] analyzed the statistical inference of dependent competing risks’ model using the Marshall–Olkin bivariate Weibull (MOBW) distribution. Bai et al. [6] considered the maximum likelihood estimations (MLEs) and the confidence intervals (CIs) for MOBW distribution under the type-I progressive interval censoring scheme. Kundu and Gupta [7] discussed the Bayes estimations (BEs) of the MOBW distribution. Feizjavadian and Hashemi [8] investigated the MOBW distribution under progressive hybrid censoring.In the competing risks’ model, the failure data of each unit is represented by a bivariate form data, which is composed of the failure time and corresponding failure cause. In the previous studies on competing risks’ models, it is usually assumed that the failure cause of unite can be observed. However, in some cases, because of the high cost or the other reasons, the determination cause of failure may be difficult to obtain [9, 10]. Thus, the failure time is observed, while no corresponding failure occurs.Censoring has become a part of the life test where the test terminates before all of the undergoing units fail. This is done to save and reduce duration of the experiment and cost incurred in the test. Among the censoring schemes, the adaptive type-II progressive hybrid censored scheme (AT-II PHCS), introduced by Ng et al. [11], is one of the popular schemes, where one can obtain effective failure numbers for the statistical inference and also speed up the test when the test reaches the given time. The AT-II PHCS can be described as follows. Suppose n identical units are placed on the life test. The observed failure number mm<n is fixed in advance, and the experiment time is allowed to run over the prefixed time τ,τ∈0,∞. The progressive censoring scheme R1,R2,…,Rm is allowed to change according to the m failure times T1:m:n,T2:m:n,…,Tm:m:n. For convenience, let Ti=Ti:m:n,i=1,2,…,m. At the time of the first failureT1, R1units are randomly removed from the remaining n−1 units. Similarly, at the second failure T2, R2 units from the remaining n−2−R1 units are randomly removed and so on. If the mth observed failure occurs before time τ (i.e., Tm<τ), the test stops at time Tm, and all remaining Rm units are removed, where Rm=n−m−∑i=1mRi. Otherwise, if TJ<τ<TJ+1, where J+1<m, and we adapt the number of units removed from the test upon failures by setting RJ+1=RJ+2=...=Rm−1=0 and Rm=n−m−∑i=1JRi. In this case, the actually applied progressive censored scheme is R1,...,RJ,0,...,0,n−m−∑i=1JRi. Presently, some scholars discussed the AT-II PHCS, i.e., Cramer and Iliopoulos [12] derived some distributional results and showed that maximum likelihood estimators coincide with those in deterministic progressive type-II censoring. Hemmati and Khorram [13] discussed the AT-II PHCS in the presence of the competing risk. Nassar and Abo-Kasem [14] adopted the frequents and Bayse methods to obtain the estimations of the unknown parameters for inverse Weibull distribution under AT-II PHCS.Different from the previous studies, the dependent competing risks’ model using Marshall–Olkin bivariate Gompertz (MOBG) with nonconstant shape parameter in constant stress accelerated life test (CSALT) is considered under AT-IIPHCS. In addition, we assume that the failure causes of some units are unknown. The Gompertz distribution is first introduced by Gompertz [15] to describe human mortality and establish actuarial tables. This distribution is a generalization of the exponential distribution and has many real life applications, especially in medical and actuarial studies. To quote a select few, Doblhammer [16], Perozek [17], and Willemse and Koppelaar [18] adopted this distribution to describe the adult lifespan by demographers and actuaries. Finch and Pike [19] and Ricklefs and Scheuerlein [20] used the Gompertz distribution to fit the mortality data of birds and mammals. In addition, this distribution which is used as a statistical model is fitted the tumor growth in survival analysis and reliability. Statistical inference for the Gompertz distribution has received many authors’ attention, such as Wu and Shi [21] considered the BEs of the Gompertz distribution in the competing risks’ model under progressively hybrid censoring. Liu et al. [22] studied the estimation for stress-strength reliability of the system with multiple types of components based on the Gompertz distribution. Mohie El-Din et al. [23] investigated the MLEs and BEs of the Gompertz distribution based on the generalized progressively hybrid censored scheme. Dey et al. [24] considered the properties and the estimation of unknown parameters of Gompertz distribution and so on.This paper aims to investigate statistical analysis of the dependent competing risk model by using the MOBG distribution under CSALT with AT-II PHCS, where the shape and scale parameters are both changed with the stress levels. The rest of this paper is organized as follows. In Section2, we describe the MOBG distribution and the accelerated dependent competing risks’ model in which the shape and scale parameters are both changed with the stress levels and the failure causes of some units are unknown. In Section 3, MLEs and BEs of the unknown parameters are obtained based on the discrete distribution and gamma distribution. The CIs of the unknown parameters based on MLEs and BEs are considered in Section 4. The simulation results and real data analysis are showed in Section 5. Conclusions are provided in Section 6.


## 2. Model Description

### 2.1. The Marshall–Olkin Gompertz Distribution

The probability density function (PDF) and the cumulative distribution function (CDF) of the Gompertz distribution with parametersλ and α can be represented as follows, respectively:(1)ft=αexpλtexp−αλexpλt−1,t>0,α>0,λ>0,(2)Ft=1−exp−αλexpλt−1,t>0,α>0,λ>0.The reliability function of Gompertz distribution can be expressed as follows:(3)Rt=exp−αλexpλt−1,t>0,α>0,λ>0,where λ is the shape parameter and α is the scale parameter. We denote the Gompertz distribution with parameters λ and α as GPλ,α.Suppose that the random variableUi,i=0,1,and2, are independent of each other and Ui∼GPλ,αi. Let X=minU0,U1 and Y=minU0,U2, then the bivariate random variable X,Y is followed by the MOBG distribution with shape parameter λ and scale parameters α0,α1, and α2, and it is denoted by MOBGλ,α0,α1,α2. Thus, the joint survival function of X,Y is shown as follows:(4)SX,Yx,y=PX>x,Y>y,=PU0>maxx,y,U1>x,U2>y,=Smaxx,y|λ,α0Sx|λ,α1Sy|λ,α2,=Sx|λ,α1Sy|λ,α0+α2,ifx<y,Sx|λ,α0+α1Sy|λ,α2,ifx>y,Sz|λ,α0+α1+α2,ifx=y=z.And, the joint PDF ofX,Y can be obtained as the following expression:(5)fX,Yx,y=f1x,y,ifx<y,f2x,y,ifx<y,f0x,y,ifx=y=z,where(6)f1x,y=fx|λ,α1fy|λ,α0+α2,f2x,y=fx|λ,α2fy|λ,α0+α1,f0z=α0α0+α1+α2fz|λ,α0+α1+α2.


### 2.2. Model Description

In this section, the dependent competing risks’ model in CSALT under AT-II PHCS is considered.LetS1<...<Sl be the accelerated stress levels and S0 be the normal stress level. The AT-II PHCS under CSALT can be run as follows. Under the ith stress level Si,i=1,2…,l, and suppose ni identical units are put on the life test, the observed failure number mimi<ni, terminal time τ, and progressive censoring scheme Ri1,Ri2,…,Rimi are given in advance. For the given τ, if TiJi<τ<TiJi+1, where Ji+1<mi, the removed units from the test upon failures is Rij∗=RijITiJi<τ,j=1,2,…,mi−1, and Rimi∗=∑j=Ji+1miRij. Otherwise, Rij∗=Rij,j=1,2,…,mi.The following basic assumptions are made for the statistical inference of the dependent competing risks’ model.A1: under the stress level Si,i=1,2,…,l; for the jth unit, there are two random variables Xij and Yij, which are the failure time under first and second failure causes, respectively. Thus, the failure time of the unit is Tij=minXij,Yij.A2: under the stress level Si,i=1,2,…,l; the pair of random variable Xij,Yij follows a MOBGλi,αi0,αi1,αi2. Xij follows GPλi,αi01, Yij follows GPλi,αi02, and Tij=minXij,Yij follows GPλi,αi where, αi01=αi0+αi1, αi02=αi0+αi2, and ai=ai0+ai1+ai2.A3: the accelerated function of the scale parameter is log-linear:(7)logαik=aik+bikφSi,i=1,2,…,l,k=0,1,and2.The accelerated function of the shape parameter is log-linear:(8)logλi=a+bφSi,i=1,2,…,l,where a,b,aik, and bik are the accelerated factors and the unknown parameters and φSi is the known function of the stress level SiBased on the model description, we can obtain the adaptive type-II progressive hybrid censored competing risks’ dataTi1,ci1,Ri1∗,Ti2,ci2,Ri2∗,…,Timi,cimi,Rimi∗, where cij∈0,1,2,∗ is failure cause of the ith unit, and it is defined as follows:(9)cij=0,ifXij=Yij,1,ifXij<Yij2,ifXij>Yij,,j=1,2,…,mi,while cij=∗ indicates that the failure cause is uncertain.Under the stress levelSi,i=1,2,…,l, let mik=∑j=1miδkcij,k=0,1,2represent the failure numbers of those caused by both risks, first risk and second risk, where δk⋅,k=0,1,and2, is the indicator function; δkcij=1,ifcij=k;δkcij=0,ifcij≠k. In addition, mi3=∑j=1mδcij=∗ represents the numbers of the failure units which have the failure time, while not having the corresponding failure cause. Let mi0+mi1+mi2=mi∗ and mi∗+mi3=mi, and we can obtain the following result.Theorem 1.
Under the stress levelSi,i=1,2,…,l,(10)mi0,mi1,mi2∼multinomialmi∗,αi0αi,αi1αi,αi2αi,where αi=αi0+αi1+αi2.Proof.
Under the stress levelSi,i=1,2,…,l, each unit failure is caused by one of Xij>Yij, Xij<Yij, or Xij=Yij. Probability of each case is(11)PXij<Yij=∫0+∞∫0yijf1xij,yijdxijdyij=αi1αi,PYij<Xij=∫0+∞∫0xijkf1xij,yijdyijdxij=αi2αi.
For the givenmi∗,mi0,mi1, and mi2, we can obtain PYij=Xij=αi0/αi by(12)∫0+∞∫0yijf1xij,yijdxijdyij+∫0+∞∫0xijf2xij,yijdyijdxij+∫0+∞f0xdx=1.
Then, the theorem is completed.


## 2.1. The Marshall–Olkin Gompertz Distribution

The probability density function (PDF) and the cumulative distribution function (CDF) of the Gompertz distribution with parametersλ and α can be represented as follows, respectively:(1)ft=αexpλtexp−αλexpλt−1,t>0,α>0,λ>0,(2)Ft=1−exp−αλexpλt−1,t>0,α>0,λ>0.The reliability function of Gompertz distribution can be expressed as follows:(3)Rt=exp−αλexpλt−1,t>0,α>0,λ>0,where λ is the shape parameter and α is the scale parameter. We denote the Gompertz distribution with parameters λ and α as GPλ,α.Suppose that the random variableUi,i=0,1,and2, are independent of each other and Ui∼GPλ,αi. Let X=minU0,U1 and Y=minU0,U2, then the bivariate random variable X,Y is followed by the MOBG distribution with shape parameter λ and scale parameters α0,α1, and α2, and it is denoted by MOBGλ,α0,α1,α2. Thus, the joint survival function of X,Y is shown as follows:(4)SX,Yx,y=PX>x,Y>y,=PU0>maxx,y,U1>x,U2>y,=Smaxx,y|λ,α0Sx|λ,α1Sy|λ,α2,=Sx|λ,α1Sy|λ,α0+α2,ifx<y,Sx|λ,α0+α1Sy|λ,α2,ifx>y,Sz|λ,α0+α1+α2,ifx=y=z.And, the joint PDF ofX,Y can be obtained as the following expression:(5)fX,Yx,y=f1x,y,ifx<y,f2x,y,ifx<y,f0x,y,ifx=y=z,where(6)f1x,y=fx|λ,α1fy|λ,α0+α2,f2x,y=fx|λ,α2fy|λ,α0+α1,f0z=α0α0+α1+α2fz|λ,α0+α1+α2.


## 2.2. Model Description

In this section, the dependent competing risks’ model in CSALT under AT-II PHCS is considered.LetS1<...<Sl be the accelerated stress levels and S0 be the normal stress level. The AT-II PHCS under CSALT can be run as follows. Under the ith stress level Si,i=1,2…,l, and suppose ni identical units are put on the life test, the observed failure number mimi<ni, terminal time τ, and progressive censoring scheme Ri1,Ri2,…,Rimi are given in advance. For the given τ, if TiJi<τ<TiJi+1, where Ji+1<mi, the removed units from the test upon failures is Rij∗=RijITiJi<τ,j=1,2,…,mi−1, and Rimi∗=∑j=Ji+1miRij. Otherwise, Rij∗=Rij,j=1,2,…,mi.The following basic assumptions are made for the statistical inference of the dependent competing risks’ model.A1: under the stress level Si,i=1,2,…,l; for the jth unit, there are two random variables Xij and Yij, which are the failure time under first and second failure causes, respectively. Thus, the failure time of the unit is Tij=minXij,Yij.A2: under the stress level Si,i=1,2,…,l; the pair of random variable Xij,Yij follows a MOBGλi,αi0,αi1,αi2. Xij follows GPλi,αi01, Yij follows GPλi,αi02, and Tij=minXij,Yij follows GPλi,αi where, αi01=αi0+αi1, αi02=αi0+αi2, and ai=ai0+ai1+ai2.A3: the accelerated function of the scale parameter is log-linear:(7)logαik=aik+bikφSi,i=1,2,…,l,k=0,1,and2.The accelerated function of the shape parameter is log-linear:(8)logλi=a+bφSi,i=1,2,…,l,where a,b,aik, and bik are the accelerated factors and the unknown parameters and φSi is the known function of the stress level SiBased on the model description, we can obtain the adaptive type-II progressive hybrid censored competing risks’ dataTi1,ci1,Ri1∗,Ti2,ci2,Ri2∗,…,Timi,cimi,Rimi∗, where cij∈0,1,2,∗ is failure cause of the ith unit, and it is defined as follows:(9)cij=0,ifXij=Yij,1,ifXij<Yij2,ifXij>Yij,,j=1,2,…,mi,while cij=∗ indicates that the failure cause is uncertain.Under the stress levelSi,i=1,2,…,l, let mik=∑j=1miδkcij,k=0,1,2represent the failure numbers of those caused by both risks, first risk and second risk, where δk⋅,k=0,1,and2, is the indicator function; δkcij=1,ifcij=k;δkcij=0,ifcij≠k. In addition, mi3=∑j=1mδcij=∗ represents the numbers of the failure units which have the failure time, while not having the corresponding failure cause. Let mi0+mi1+mi2=mi∗ and mi∗+mi3=mi, and we can obtain the following result.Theorem 1.
Under the stress levelSi,i=1,2,…,l,(10)mi0,mi1,mi2∼multinomialmi∗,αi0αi,αi1αi,αi2αi,where αi=αi0+αi1+αi2.Proof.
Under the stress levelSi,i=1,2,…,l, each unit failure is caused by one of Xij>Yij, Xij<Yij, or Xij=Yij. Probability of each case is(11)PXij<Yij=∫0+∞∫0yijf1xij,yijdxijdyij=αi1αi,PYij<Xij=∫0+∞∫0xijkf1xij,yijdyijdxij=αi2αi.
For the givenmi∗,mi0,mi1, and mi2, we can obtain PYij=Xij=αi0/αi by(12)∫0+∞∫0yijf1xij,yijdxijdyij+∫0+∞∫0xijf2xij,yijdyijdxij+∫0+∞f0xdx=1.
Then, the theorem is completed.


## 3. Point Estimations

### 3.1. Maximum Likelihood Estimations

LetT=Tij,i=1,2,...,l,j=1,2,...,mi is the censored sample in CSALT and t=tij,i=1,2,…,l,j=1,2,…,mi is the sample observation. Let θ=λ1,…,λl,a10,a11,a12,…,al0,al1,al2; under the stress level Si,i=1,2,…,l, based on the dependent competing risks’ data, the likelihood function can be expressed as follows:(13)lit|λi,αi0,αi1,αi2∝∏j=1mif0tij,tijδ0cij−SX,Yx,y∂x|tij,tijδ1cij−SX,Yx,y∂ytij,tij|δ2cijf0tij,tij+−SX,Yx,y∂x|tij,tij+−SX,Yx,y∂y|tij,tijδcij=∗SX,Yx,y|tij,tijRij∗,where(14)f0tij,tij=αi0expλitijexp−αi0+αi1+αi2λiexpλitij−1,−SX,Yx,y∂x|tij,tij=αi1expλitijexp−αi0+αi1+αi2λiexpλitij−1,−SX,Yx,y∂y|tij,tij=αi2expλitijexp−αi0+αi1+αi2λiexpλtij−1,−SX,Yx,y|tij,tij=exp−αi0+αi1+αi2λiexpλitij−1.So, we can obtain(15)lit|λ,αi0,αi1,αi2∝αi0mi0αi1mi1αi2mi2αimi−mi∗expλi∑j=1mitij−λi−1αiAi,where Ai=∑j=1miRij∗+1expλitij−1.The full likelihood function is(16)L=∏i=1lli.Based on equation (16), the log-likelihood function can be transformed into the following result:(17)lnL=∑i=1l∑k=02miklnαik+mi−mi∗lnαi+λi∑j=1mitij−λi−1αiAi.The MLEs of unknown parameters can be obtained by the maximum function in equation (17). By setting the first partial derivative of lnL with respect to λiandαik,k=0,1,and2, to zero, we can obtain following results:(18)∂lnL∂αik=mikαik+mi−mi∗αi0+αi1+αi2−1λiAi=0,k=0,1,and2,(19)∂lnL∂λi=∑j=1mitij+λi−2αiAi−λi−1αi∑j=1mitijRij∗+1expλitij=0.By solving equation (18), we can get the MLE of αik,k=0,1,and2, as the following form:(20)α^ik=mikmiλimi∗Ai,k=0,1,and2.Substitutingα^ik,k=0,1,and2, into equation (19), we can obtain the equation for λi:(21)Hλi=∑j=1mitij+λi−1mi−mi∑j=1mitijRij∗+1expλitijAi=0.Theorem 2.
Based on the censored datati1≤ti2≤...≤tij≤...≤timiand the progressive scheme Rij∗,j=1,2,…,mi, the MLEs λ^i and α^ik,k=0,1,and2, of the parameters λi and αik,k=0,1,and2, exist and are unique if and only if(22)2∑j=1mitij∑j=1miRij∗+1tij>mi∑j=1miRij∗+1tij2.Proof.
From equations (20) and (21), we can obtain that λ^i and α^ik,k=0,1,and2, exist and are unique by showing that λ^i exists and is unique if and only if condition (22) holds. The derivative of Hλi can be expressed as follows:(23)H′λi=−miλi2−mi∑j=1miRij∗+1expλitijtij2∑j=1miRij∗+1expλitij−1−mi∑j=1miRij∗+1expλitijtij2∑j=1miRij∗+1expλitij−12.
Letzj=zjλi=λitij, we have(24)H′λi=−miλi21+∑j=1miRij∗+1expzjzj2∑j=1miRij∗+1expzj−1−∑j=1miRij∗+1expzjzj2∑j=1miRij∗+1expzj−12,=−miλi2H1λi∑j=1miRij∗+1expzj−1+H2λi∑j=1miRij∗+1expzj−12,where(25)H1λi=∑j=1miRij∗+11expzj−1expzj−12−zi2expzj,H2λi=∑j=1miRij∗+1expzj−1∑j=1miRij∗+1ziexpzj2expzj−1−∑j=1miRij∗+1expzjzj2.
Sinceexpzj−12>zi2expzj for all zj>0, we have H1λi>0.
Now, let(26)hj=Rij∗+1expzj−11/2,gj=Rij∗+1expzj−11/2ziexpzj.
Then, we have(27)H2λi=∑j=1mihj2∑j=1migj2−∑j=1mihjgj2.
By using the Cauchy–Schwarz inequality, we can obtainH2λi≥0. Then, it follows that H′λi<0 which implies that Hλi is a decrease function. Therefore, equation (21) has the unique root λ^i∈0,∞ if and only if H0>0 and H∞<0:(28)H0=∑j=1mitij+milimλi⟶01λi−∑j=1mitijRij∗+1expλitij∑j=1miRij∗+1expλitij−1,=∑j=1mitij−mi2∑j=1miRij∗+1tij2∑j=1miRij∗+1tij,=2∑j=1mitij∑j=1miRij∗+1tij−mi∑j=1miRij∗+1tij22∑j=1miRij∗+1tij.
If and only if equation (22) is satisfied, H0>0:(29)H∞=∑j=1mitij−milimλi⟶∞∑j=1mitijRij∗+1expλitij∑j=1miRij∗+1expλitij−1,=∑j=1mitij−milimλi⟶∞∑j=1mitij2Rij∗+1expλitij−timi∑j=1mitijRij∗+1expλitij−timi,=∑j=1mitij−timi<0.
Then, the theorem is completed.Theorem 3.
Using the MLE of the parameterα^ik,k=0,1,and2, we can get the MLE ϑ^=α^i0+α^i1+α^i2 of the parameter ϑ=αi0+αi1+αi2 having the following properties:(1)
ϑ^ is the asymptotic unbiased estimator of ϑ(2)
ϑ^ is a consistent estimator of ϑ(3)
ϑ^ is a sufficient statistics of ϑProof.
From A2,Tij follows the Gompertz distribution with the shape parameter λi and scale parameter αi. By the variable transformed method, the random variable Yij=expλiTij−1/λi follows the exponential distribution with mean 1/αi.
Let(30)Zi1=niYi1,Zi2=ni−Ri1∗−1Yi2−Yi1,…Zimi=ni−Ri1∗−…Rimi∗−mi−1Yimi−Yi,mi−1.
We can obtain thatZi1,Zi2,…,Zimi are independent and identical distribution as the exponential distribution with mean 1/αi. According to the relationship between the exponential distribution and gamma distribution, we have(31)Zij∼Eαi=αie−αizij=αi1Γ1zij1−1e−αizij∼Gamma1,αi.
LetWi=Zi1+Zi2+⋯Zimi, then Wi∼Gammami,αi:(32)Wi=Zi1+Zi2+⋯Zimi=∑j=1miRij∗+1expλitij−1/λ=Aiλi.
Then,λi/Ai follows the inverse gamma distribution with the parameters miand αi, and it is denoted by λi/Ai∼IGami,αi.
Using equation (20),(33)Eα^i0+α^i1+α^i2=EmiλiAi=miαi0+αi1+αi2mi−1.
Then, we have(34)limmi⟶∞Eα^i0+α^i1+α^i2=αi0+αi1+αi2.
The proof of statement 1 is completed.
Using equation (20), we can obtain(35)Varα^i0+α^i1+α^i2=EmiλiAi=mi2αi0+αi1+αi22mi−1mi−2.
Then,(36)limmi⟶∞Varα^i0+α^i1+α^i2=αi0+αi1+αi22limmi⟶∞mi2mi−12mi−2=0.
The proof of statement 2 is completed.
Finally, under the stress levelSi,i=1,2,…,l, the likelihood function can be written as follows:(37)li∝αi0mi0αi1mi1αi2mi2αimi−mi∗expλi∑j=1mitij−λi−1αi0+αi1+αi2Ai.
Based on equation (20), we can obtain(38)Ai=λimiα^i0+α^i1+α^i2.
Then, the likelihood function can be written as follows:(39)li∝αi0mi0αi1mi1αi2mi2αimi−mi∗expλi∑j=1mitij−miαi0+αi1+αi2/α^i0+α^i1+α^i2∝Gti1,…,timiHαi0,αi1,αi2,α^i0,α^i1,α^i2.
The likelihood function can be decomposed into two functionsGti1,…,timi and Hαi0,αi1,αi2,α^i0,α^i1,α^i2, where G is only dependent on the data and H is dependent on the parameters. This completes the proof of statement 3 and the theorem.


### 3.2. The Bayes Estimations

We assume that the parametersλi,αi0,αi1, and αi2 are independent of each other. According to Wu and Shi [21], for the parameter λi, we select a finite number of values on the interval 0,∞, which is denoted by λi1,λi2,…,λir. The PDF of the parameters λi is πλi=λir=πir, where ∑r=1qπir=1,0≤πir≤1,i=1,2,…,l.Based on equation (16), we can obtain the conditional probability density function of parameter αik,k=0,1,and2:(40)pαik|λ∝αikmikαimi−mi∗expλi−1αikAi∝∑q=1mi−mi∗αikmik+qexpλi−1αikAi.Whenmi−mi∗=0, the conditional probability density function of parameter αik,k=0,1,and2, can be expressed as follows:(41)pαik|λ∝αikmikexpλi−1αikAi.Then, we can obtain thatpαik|λ belongs to the gamma distribution family.Under the conditionλi=λir, the condition prior distribution of the parameter αik,k=0,1,and2, is Gammaaikr,bikr, which is the conjugate distribution for the parameter αik,k=0,1,and2. The PDF can be expressed as follows:(42)παik|λi=λir=dirkcirkαikairk−1exp−dikrαikΓcirk.The condition posterior density function of the parameterαik,k=0,1,and2, is(43)παik|t,λi=λir=liθπαik|λi=λirπλi=λir∫liθπαik|λi=λirπλi=λirdαik.Using the binomial expansion,(44)αi0+αi1+αi2mi−mi∗=∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvαi0uαi1vαi2mi−mi∗−u−v.The likelihood function in equation (15) can be written as the following form:(45)lit|λ,αi0,αi1,αi2∝∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvαi0mi0+uαi1mi1+vαi2mi2+mi−mi∗−u−vexpλi∑j=1mitij−λi−1αiAi.Then, the condition posterior density function of the parameterαik,k=0,1,and2, can be obtained as follows:(46)παi0|λi=λir,t=Bir−1exp−αi0λirAir+dir0×∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−vmi−mi∗−uvΓmi1+cir1+vλirAir+dir1mi1+cir1+v×Γmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+cir2+mi−mi∗−u−vαi0mi0+cir0+u−1,(47)παi1|λi=λir,t=Bir−1exp−αi1λirAir+dir1×∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+uλirAir+dir0mi0+ai0+u×Γmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+ai2+mi−mi∗−u−vαi1mi0+cir0+v−1,(48)παi2|t,λi=λir=Bir−1exp−αi2λirAir+bir2×∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+air0+uλirAir+bir0mi0+air0+u×Γmi1+air1+vλirAir+bir1mi1+air1+vαi2mi2+air2+mi−mi∗−u−v,where Air=∑j=1miRij∗+1expλirtij−1.The marginal posterior density distribution of the parameterλi,i=1,2,…,l, can be obtained as follows:(49)Pir=Pλi=λir=∫∫∫∏j=1milit|λi,αi0,αi1,αi2παi0,αi1,αi2|λi=λirπλi=λirdαi0dαi1dαi2∑r=1qi∫∫∫∏j=1milit|λi,αi0,αi1,αi2παi0,αi1,αi2|λi=λirπλi=λirdαi0dαi1dαi2,=πir∏i=1l∏k=02dirkairk/ΓcirkBir∑r=1qiπir∏i=1l∏k=02dirkairk/ΓcirkBir,where(50)Bir=∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+uλirAir+dir0mi0+air0+u×Γmi1+cir1+vλirAir+dir1mi1+cir1+vΓmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+cir2+mi−mi∗−u−v.Under the squared error loss function, the BEs ofλi,αi0,αi1, and αi2 can be written as follows:(51)λ^iBS=Eλi|tλi=∑r=1qiPirλir,α^i0BS=Eαi0|λ,tαi0=∑r=1qiPirEαi0=∑r=1qiPir1Bir∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+u+1λirAri+dir0mi0+cir0+u+1×Γmi1+cir1+vλirAri+dir1mi1+cir1+vΓmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+cir2+mi−mi∗−u−v,α^i1BS=Eαi1|λ,tαi1=∑r=1qiPirEαi1=∑r=1qiPir1Bir∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+uλirAri+dir0mi0+cir0+u×Γmi1+cir1+v+1λirAri+dir1mi1+cir1+v+1Γmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+cir2+mi−mi∗−u−v,α^i2BS=Eαi2|λ,tαi2=∑r=1qiPirEαi2=∑r=1qiPir1Bir∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+uλirAri+dir0mi0+cir0+u×Γmi1+cir1+vλirAri+dir1mi1+cir1+vΓmi2+cir2+mi−mi∗−u−v+1λirAir+dir2mi2+cir2+mi−mi∗−u−v+1.


### 3.3. The Estimation of Reliability

The reliability estimation of the unit at any timet under the normal stress level is(52)R^t=exp−α^00+α^01+α^02λ^0expλ^0t−1.The estimation of the shape parameterλ0 under normal stress level S0 can be obtained from the equation logλ0=a+bφS0,i=1,2,…,l. And, the estimation of the scale parameter α0k under normal stress level S0 can be obtained from the equation logα0k=a0k+b0kφS0,i=1,2,…,l,k=0,1,and2. Putting the estimations of the parameters λ^iandα^ik,i=1,2,…,l,k=0,1,and2, into equations (5) and (6), we can obtain the estimations of a,b,a0k,andb0k,k=0,1,and2,by the least square method:(53)a^=∑i=1llnλ^i−b^∑i=1lφSil,b^=l∑i=1llnλ^iφSi−∑i=1llnλ^i∑i=1lφSil∑i=1lφ2Si−∑i=1lφSi2,a^ik=∑i=1llnα^ik−b^ik∑i=1lφSil,b^ik=l∑i=1lφSilnα^ik−∑i=1llnα^ik∑i=1lφSil∑i=1lφ2Si−∑i=1lφSi2.


## 3.1. Maximum Likelihood Estimations

LetT=Tij,i=1,2,...,l,j=1,2,...,mi is the censored sample in CSALT and t=tij,i=1,2,…,l,j=1,2,…,mi is the sample observation. Let θ=λ1,…,λl,a10,a11,a12,…,al0,al1,al2; under the stress level Si,i=1,2,…,l, based on the dependent competing risks’ data, the likelihood function can be expressed as follows:(13)lit|λi,αi0,αi1,αi2∝∏j=1mif0tij,tijδ0cij−SX,Yx,y∂x|tij,tijδ1cij−SX,Yx,y∂ytij,tij|δ2cijf0tij,tij+−SX,Yx,y∂x|tij,tij+−SX,Yx,y∂y|tij,tijδcij=∗SX,Yx,y|tij,tijRij∗,where(14)f0tij,tij=αi0expλitijexp−αi0+αi1+αi2λiexpλitij−1,−SX,Yx,y∂x|tij,tij=αi1expλitijexp−αi0+αi1+αi2λiexpλitij−1,−SX,Yx,y∂y|tij,tij=αi2expλitijexp−αi0+αi1+αi2λiexpλtij−1,−SX,Yx,y|tij,tij=exp−αi0+αi1+αi2λiexpλitij−1.So, we can obtain(15)lit|λ,αi0,αi1,αi2∝αi0mi0αi1mi1αi2mi2αimi−mi∗expλi∑j=1mitij−λi−1αiAi,where Ai=∑j=1miRij∗+1expλitij−1.The full likelihood function is(16)L=∏i=1lli.Based on equation (16), the log-likelihood function can be transformed into the following result:(17)lnL=∑i=1l∑k=02miklnαik+mi−mi∗lnαi+λi∑j=1mitij−λi−1αiAi.The MLEs of unknown parameters can be obtained by the maximum function in equation (17). By setting the first partial derivative of lnL with respect to λiandαik,k=0,1,and2, to zero, we can obtain following results:(18)∂lnL∂αik=mikαik+mi−mi∗αi0+αi1+αi2−1λiAi=0,k=0,1,and2,(19)∂lnL∂λi=∑j=1mitij+λi−2αiAi−λi−1αi∑j=1mitijRij∗+1expλitij=0.By solving equation (18), we can get the MLE of αik,k=0,1,and2, as the following form:(20)α^ik=mikmiλimi∗Ai,k=0,1,and2.Substitutingα^ik,k=0,1,and2, into equation (19), we can obtain the equation for λi:(21)Hλi=∑j=1mitij+λi−1mi−mi∑j=1mitijRij∗+1expλitijAi=0.Theorem 2.
Based on the censored datati1≤ti2≤...≤tij≤...≤timiand the progressive scheme Rij∗,j=1,2,…,mi, the MLEs λ^i and α^ik,k=0,1,and2, of the parameters λi and αik,k=0,1,and2, exist and are unique if and only if(22)2∑j=1mitij∑j=1miRij∗+1tij>mi∑j=1miRij∗+1tij2.Proof.
From equations (20) and (21), we can obtain that λ^i and α^ik,k=0,1,and2, exist and are unique by showing that λ^i exists and is unique if and only if condition (22) holds. The derivative of Hλi can be expressed as follows:(23)H′λi=−miλi2−mi∑j=1miRij∗+1expλitijtij2∑j=1miRij∗+1expλitij−1−mi∑j=1miRij∗+1expλitijtij2∑j=1miRij∗+1expλitij−12.
Letzj=zjλi=λitij, we have(24)H′λi=−miλi21+∑j=1miRij∗+1expzjzj2∑j=1miRij∗+1expzj−1−∑j=1miRij∗+1expzjzj2∑j=1miRij∗+1expzj−12,=−miλi2H1λi∑j=1miRij∗+1expzj−1+H2λi∑j=1miRij∗+1expzj−12,where(25)H1λi=∑j=1miRij∗+11expzj−1expzj−12−zi2expzj,H2λi=∑j=1miRij∗+1expzj−1∑j=1miRij∗+1ziexpzj2expzj−1−∑j=1miRij∗+1expzjzj2.
Sinceexpzj−12>zi2expzj for all zj>0, we have H1λi>0.
Now, let(26)hj=Rij∗+1expzj−11/2,gj=Rij∗+1expzj−11/2ziexpzj.
Then, we have(27)H2λi=∑j=1mihj2∑j=1migj2−∑j=1mihjgj2.
By using the Cauchy–Schwarz inequality, we can obtainH2λi≥0. Then, it follows that H′λi<0 which implies that Hλi is a decrease function. Therefore, equation (21) has the unique root λ^i∈0,∞ if and only if H0>0 and H∞<0:(28)H0=∑j=1mitij+milimλi⟶01λi−∑j=1mitijRij∗+1expλitij∑j=1miRij∗+1expλitij−1,=∑j=1mitij−mi2∑j=1miRij∗+1tij2∑j=1miRij∗+1tij,=2∑j=1mitij∑j=1miRij∗+1tij−mi∑j=1miRij∗+1tij22∑j=1miRij∗+1tij.
If and only if equation (22) is satisfied, H0>0:(29)H∞=∑j=1mitij−milimλi⟶∞∑j=1mitijRij∗+1expλitij∑j=1miRij∗+1expλitij−1,=∑j=1mitij−milimλi⟶∞∑j=1mitij2Rij∗+1expλitij−timi∑j=1mitijRij∗+1expλitij−timi,=∑j=1mitij−timi<0.
Then, the theorem is completed.Theorem 3.
Using the MLE of the parameterα^ik,k=0,1,and2, we can get the MLE ϑ^=α^i0+α^i1+α^i2 of the parameter ϑ=αi0+αi1+αi2 having the following properties:(1)
ϑ^ is the asymptotic unbiased estimator of ϑ(2)
ϑ^ is a consistent estimator of ϑ(3)
ϑ^ is a sufficient statistics of ϑProof.
From A2,Tij follows the Gompertz distribution with the shape parameter λi and scale parameter αi. By the variable transformed method, the random variable Yij=expλiTij−1/λi follows the exponential distribution with mean 1/αi.
Let(30)Zi1=niYi1,Zi2=ni−Ri1∗−1Yi2−Yi1,…Zimi=ni−Ri1∗−…Rimi∗−mi−1Yimi−Yi,mi−1.
We can obtain thatZi1,Zi2,…,Zimi are independent and identical distribution as the exponential distribution with mean 1/αi. According to the relationship between the exponential distribution and gamma distribution, we have(31)Zij∼Eαi=αie−αizij=αi1Γ1zij1−1e−αizij∼Gamma1,αi.
LetWi=Zi1+Zi2+⋯Zimi, then Wi∼Gammami,αi:(32)Wi=Zi1+Zi2+⋯Zimi=∑j=1miRij∗+1expλitij−1/λ=Aiλi.
Then,λi/Ai follows the inverse gamma distribution with the parameters miand αi, and it is denoted by λi/Ai∼IGami,αi.
Using equation (20),(33)Eα^i0+α^i1+α^i2=EmiλiAi=miαi0+αi1+αi2mi−1.
Then, we have(34)limmi⟶∞Eα^i0+α^i1+α^i2=αi0+αi1+αi2.
The proof of statement 1 is completed.
Using equation (20), we can obtain(35)Varα^i0+α^i1+α^i2=EmiλiAi=mi2αi0+αi1+αi22mi−1mi−2.
Then,(36)limmi⟶∞Varα^i0+α^i1+α^i2=αi0+αi1+αi22limmi⟶∞mi2mi−12mi−2=0.
The proof of statement 2 is completed.
Finally, under the stress levelSi,i=1,2,…,l, the likelihood function can be written as follows:(37)li∝αi0mi0αi1mi1αi2mi2αimi−mi∗expλi∑j=1mitij−λi−1αi0+αi1+αi2Ai.
Based on equation (20), we can obtain(38)Ai=λimiα^i0+α^i1+α^i2.
Then, the likelihood function can be written as follows:(39)li∝αi0mi0αi1mi1αi2mi2αimi−mi∗expλi∑j=1mitij−miαi0+αi1+αi2/α^i0+α^i1+α^i2∝Gti1,…,timiHαi0,αi1,αi2,α^i0,α^i1,α^i2.
The likelihood function can be decomposed into two functionsGti1,…,timi and Hαi0,αi1,αi2,α^i0,α^i1,α^i2, where G is only dependent on the data and H is dependent on the parameters. This completes the proof of statement 3 and the theorem.


## 3.2. The Bayes Estimations

We assume that the parametersλi,αi0,αi1, and αi2 are independent of each other. According to Wu and Shi [21], for the parameter λi, we select a finite number of values on the interval 0,∞, which is denoted by λi1,λi2,…,λir. The PDF of the parameters λi is πλi=λir=πir, where ∑r=1qπir=1,0≤πir≤1,i=1,2,…,l.Based on equation (16), we can obtain the conditional probability density function of parameter αik,k=0,1,and2:(40)pαik|λ∝αikmikαimi−mi∗expλi−1αikAi∝∑q=1mi−mi∗αikmik+qexpλi−1αikAi.Whenmi−mi∗=0, the conditional probability density function of parameter αik,k=0,1,and2, can be expressed as follows:(41)pαik|λ∝αikmikexpλi−1αikAi.Then, we can obtain thatpαik|λ belongs to the gamma distribution family.Under the conditionλi=λir, the condition prior distribution of the parameter αik,k=0,1,and2, is Gammaaikr,bikr, which is the conjugate distribution for the parameter αik,k=0,1,and2. The PDF can be expressed as follows:(42)παik|λi=λir=dirkcirkαikairk−1exp−dikrαikΓcirk.The condition posterior density function of the parameterαik,k=0,1,and2, is(43)παik|t,λi=λir=liθπαik|λi=λirπλi=λir∫liθπαik|λi=λirπλi=λirdαik.Using the binomial expansion,(44)αi0+αi1+αi2mi−mi∗=∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvαi0uαi1vαi2mi−mi∗−u−v.The likelihood function in equation (15) can be written as the following form:(45)lit|λ,αi0,αi1,αi2∝∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvαi0mi0+uαi1mi1+vαi2mi2+mi−mi∗−u−vexpλi∑j=1mitij−λi−1αiAi.Then, the condition posterior density function of the parameterαik,k=0,1,and2, can be obtained as follows:(46)παi0|λi=λir,t=Bir−1exp−αi0λirAir+dir0×∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−vmi−mi∗−uvΓmi1+cir1+vλirAir+dir1mi1+cir1+v×Γmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+cir2+mi−mi∗−u−vαi0mi0+cir0+u−1,(47)παi1|λi=λir,t=Bir−1exp−αi1λirAir+dir1×∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+uλirAir+dir0mi0+ai0+u×Γmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+ai2+mi−mi∗−u−vαi1mi0+cir0+v−1,(48)παi2|t,λi=λir=Bir−1exp−αi2λirAir+bir2×∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+air0+uλirAir+bir0mi0+air0+u×Γmi1+air1+vλirAir+bir1mi1+air1+vαi2mi2+air2+mi−mi∗−u−v,where Air=∑j=1miRij∗+1expλirtij−1.The marginal posterior density distribution of the parameterλi,i=1,2,…,l, can be obtained as follows:(49)Pir=Pλi=λir=∫∫∫∏j=1milit|λi,αi0,αi1,αi2παi0,αi1,αi2|λi=λirπλi=λirdαi0dαi1dαi2∑r=1qi∫∫∫∏j=1milit|λi,αi0,αi1,αi2παi0,αi1,αi2|λi=λirπλi=λirdαi0dαi1dαi2,=πir∏i=1l∏k=02dirkairk/ΓcirkBir∑r=1qiπir∏i=1l∏k=02dirkairk/ΓcirkBir,where(50)Bir=∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+uλirAir+dir0mi0+air0+u×Γmi1+cir1+vλirAir+dir1mi1+cir1+vΓmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+cir2+mi−mi∗−u−v.Under the squared error loss function, the BEs ofλi,αi0,αi1, and αi2 can be written as follows:(51)λ^iBS=Eλi|tλi=∑r=1qiPirλir,α^i0BS=Eαi0|λ,tαi0=∑r=1qiPirEαi0=∑r=1qiPir1Bir∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+u+1λirAri+dir0mi0+cir0+u+1×Γmi1+cir1+vλirAri+dir1mi1+cir1+vΓmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+cir2+mi−mi∗−u−v,α^i1BS=Eαi1|λ,tαi1=∑r=1qiPirEαi1=∑r=1qiPir1Bir∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+uλirAri+dir0mi0+cir0+u×Γmi1+cir1+v+1λirAri+dir1mi1+cir1+v+1Γmi2+cir2+mi−mi∗−u−vλirAir+dir2mi2+cir2+mi−mi∗−u−v,α^i2BS=Eαi2|λ,tαi2=∑r=1qiPirEαi2=∑r=1qiPir1Bir∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi0+cir0+uλirAri+dir0mi0+cir0+u×Γmi1+cir1+vλirAri+dir1mi1+cir1+vΓmi2+cir2+mi−mi∗−u−v+1λirAir+dir2mi2+cir2+mi−mi∗−u−v+1.


## 3.3. The Estimation of Reliability

The reliability estimation of the unit at any timet under the normal stress level is(52)R^t=exp−α^00+α^01+α^02λ^0expλ^0t−1.The estimation of the shape parameterλ0 under normal stress level S0 can be obtained from the equation logλ0=a+bφS0,i=1,2,…,l. And, the estimation of the scale parameter α0k under normal stress level S0 can be obtained from the equation logα0k=a0k+b0kφS0,i=1,2,…,l,k=0,1,and2. Putting the estimations of the parameters λ^iandα^ik,i=1,2,…,l,k=0,1,and2, into equations (5) and (6), we can obtain the estimations of a,b,a0k,andb0k,k=0,1,and2,by the least square method:(53)a^=∑i=1llnλ^i−b^∑i=1lφSil,b^=l∑i=1llnλ^iφSi−∑i=1llnλ^i∑i=1lφSil∑i=1lφ2Si−∑i=1lφSi2,a^ik=∑i=1llnα^ik−b^ik∑i=1lφSil,b^ik=l∑i=1lφSilnα^ik−∑i=1llnα^ik∑i=1lφSil∑i=1lφ2Si−∑i=1lφSi2.


## 4. Interval Estimation

### 4.1. Approximation Confidence Intervals

For the parametersλi and αik,i=1,2,…,l,k=0,1,and2, it is impossible to get the exact confidence intervals. So, we use the asymptotic property of the MLEs to construct the approximation confidence intervals (ACIs), and the observed Fisher information matrix of the parametersλi and αik,i=1,2,…,l,k=0,1,and2is denoted by(54)I^θ=diagI^1,I^2,…,I^l,where the submatrix can be expressed as follows:(55)I^iλi,αi0,αi1,αi2=I^11iI^12iI^13iI^14iI^21iI^22iI^23iI^24iI^31iI^32iI^33iI^34iI^41iI^42iI^43iI^44i,where(56)I^11i=−∂2lnL∂λi2|λi=λ^i,α=α^i=−λ^i−1α^i∑j=1miRij∗+1expλ^itijtij2+2λ^i−2α^i∑j=1miRij∗+1expλ^itijtij−2λ^i−3α^i∑j=1miRij∗+1expλ^itij−1,I^1k+2i=I^k+21i=−∂2lnL∂λi∂αik|λi=λ^,αik=α^ik=−λ^i−1∑j=1miRij∗+1expλ^itijtij−λ^i−2∑j=1miRij∗+1expλ^itij−1,I^k+2k+2i=−∂2lnL∂αik2|αik=α^ik=−mika^ik2+−mi−mi∗a^i2,I^23i=I^32i=I^24i=I^42i=I^34i=I^43i=−∂2lnL∂αik∂αih|αik=α^ik,αih=α^ih=−mi−mi∗α^i2,kandh=0,1,and2,k≠h.The variance-covariance matrix of the parametersλi and αik,i=1,2,…,l,k=0,1,and2, can be obtained as follows:(57)V^θ=I^θ−1=diagI^1−1,I^2−1,…,I^l−1.Let(58)V^iλi,αi0,αi1,αi2=I^iλi,αi0,αi1,αi2−1=V^11iV^12iV^13iV^14iV^21iV^22iV^23iV^24iV^31iV^32iV^33iV^34iV^41iV^42iV^43iV^44i=I^11iI^12iI^13iI^14iI^21iI^22iI^23iI^24iI^31iI^32iI^33iI^34iI^41iI^42iI^43iI^44i−1.For the given confidence levelγ, the 1001−γ% ACIs of parameters λiandαik,k=0,1,and2, can be expressed as follows:(59)λ^iL,λ^iU=λ^i−Zγ/2V^11i,λ^i+Zγ/2V^11i,α^ikL,α^ikU=α^ik−Zγ/2V^k+1k+1i,α^ik+Zγ/2V^k+1k+1i,k=0,1,2,where Zγ/2 is the upper γ/2 percentile point of the standard normal distribution.


### 4.2. Highest Posterior Density Credible Intervals

The1001−γ% two-side Bayes credible interval (BCI) of αi0 which is denoted by α^i0BL,α^i0BU can be obtained by solving the following two equations:(60)γ2=∫0α^i0BLπαi0|λ,tdαi0,(61)1−γ2=∫0α^i0BUπαi0|λ,tdαi0.Substituting equation (46) into (60) and (61), we have(62)γ/2=1Bi∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi1+ci1+v×Γmi2+ci2+mi−mi∗−u−vCuvλiAi+di0−mi0+ci0+u×Γmi0+ci0+u,λiAi+di0α^i0BL,(63)1−γ2=1Bi∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi1+ci1+v×Γmi2+ci2+mi−mi∗−u−vCuvλiAi+di0−mi0+ci0+u×Γmi0+ci0+u,λiAi+di0α^i0BU,where(64)Cuv=λiAi+di0mi0+ci0+uλiAi+di1mi1+ci1+vλiAi+di2mi2+ci2+mi−mi∗−u−v.And,Γa,x=∫0xta−1e−tdt is an incomplete gamma function.Equations (62) and (63) cannot yield the explicit solutions for α^i0BL and α^i0BU. Thus, we can only obtain the numerical solution. Similarly, we can get α^ikBL,α^ikBU,k=1and2.Whenmi−mi∗=0, the parameter αik,k=0,1,and2, follows the gamma distribution with parameters mik+cik and λiAi+dik, and it is denoted by αik∼Gammamik+cik,λiAi+dik,k=0,1,and2. The 1001−γ% two-side BCI of αik,k=0,1,and2, can be acquired by the following equations:(65)γ2=∫0α^ikBUλiAi+dikmik+cikΓmik+cikαikmik+cik−1exp−αikλiAi+dikdαik,1−γ2=∫0α^ikBUλiAi+dikmik+cikΓmik+cikαikmik+cik−1exp−αikλiAi+dikdαik.In this case, ifαik,k=0,1,and2, is an integer, the 1001−γ% BCI is(66)χ1−γ/222mik+cik2λiAi+dik,χγ/222mik+cik2λiAi+dik.The highest posterior density (HPD) CIs ofαik,k=0,1,and2, say α^ikHL,α^ikHU, is the shortest interval and can be obtained from α^ikHL−α^ikHU=minα^ikBL−α^ikBU,k=0,1,and2.


## 4.1. Approximation Confidence Intervals

For the parametersλi and αik,i=1,2,…,l,k=0,1,and2, it is impossible to get the exact confidence intervals. So, we use the asymptotic property of the MLEs to construct the approximation confidence intervals (ACIs), and the observed Fisher information matrix of the parametersλi and αik,i=1,2,…,l,k=0,1,and2is denoted by(54)I^θ=diagI^1,I^2,…,I^l,where the submatrix can be expressed as follows:(55)I^iλi,αi0,αi1,αi2=I^11iI^12iI^13iI^14iI^21iI^22iI^23iI^24iI^31iI^32iI^33iI^34iI^41iI^42iI^43iI^44i,where(56)I^11i=−∂2lnL∂λi2|λi=λ^i,α=α^i=−λ^i−1α^i∑j=1miRij∗+1expλ^itijtij2+2λ^i−2α^i∑j=1miRij∗+1expλ^itijtij−2λ^i−3α^i∑j=1miRij∗+1expλ^itij−1,I^1k+2i=I^k+21i=−∂2lnL∂λi∂αik|λi=λ^,αik=α^ik=−λ^i−1∑j=1miRij∗+1expλ^itijtij−λ^i−2∑j=1miRij∗+1expλ^itij−1,I^k+2k+2i=−∂2lnL∂αik2|αik=α^ik=−mika^ik2+−mi−mi∗a^i2,I^23i=I^32i=I^24i=I^42i=I^34i=I^43i=−∂2lnL∂αik∂αih|αik=α^ik,αih=α^ih=−mi−mi∗α^i2,kandh=0,1,and2,k≠h.The variance-covariance matrix of the parametersλi and αik,i=1,2,…,l,k=0,1,and2, can be obtained as follows:(57)V^θ=I^θ−1=diagI^1−1,I^2−1,…,I^l−1.Let(58)V^iλi,αi0,αi1,αi2=I^iλi,αi0,αi1,αi2−1=V^11iV^12iV^13iV^14iV^21iV^22iV^23iV^24iV^31iV^32iV^33iV^34iV^41iV^42iV^43iV^44i=I^11iI^12iI^13iI^14iI^21iI^22iI^23iI^24iI^31iI^32iI^33iI^34iI^41iI^42iI^43iI^44i−1.For the given confidence levelγ, the 1001−γ% ACIs of parameters λiandαik,k=0,1,and2, can be expressed as follows:(59)λ^iL,λ^iU=λ^i−Zγ/2V^11i,λ^i+Zγ/2V^11i,α^ikL,α^ikU=α^ik−Zγ/2V^k+1k+1i,α^ik+Zγ/2V^k+1k+1i,k=0,1,2,where Zγ/2 is the upper γ/2 percentile point of the standard normal distribution.


## 4.2. Highest Posterior Density Credible Intervals

The1001−γ% two-side Bayes credible interval (BCI) of αi0 which is denoted by α^i0BL,α^i0BU can be obtained by solving the following two equations:(60)γ2=∫0α^i0BLπαi0|λ,tdαi0,(61)1−γ2=∫0α^i0BUπαi0|λ,tdαi0.Substituting equation (46) into (60) and (61), we have(62)γ/2=1Bi∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi1+ci1+v×Γmi2+ci2+mi−mi∗−u−vCuvλiAi+di0−mi0+ci0+u×Γmi0+ci0+u,λiAi+di0α^i0BL,(63)1−γ2=1Bi∑u=0mi−mi∗mi−mi∗u∑v=0mi−mi∗−umi−mi∗−uvΓmi1+ci1+v×Γmi2+ci2+mi−mi∗−u−vCuvλiAi+di0−mi0+ci0+u×Γmi0+ci0+u,λiAi+di0α^i0BU,where(64)Cuv=λiAi+di0mi0+ci0+uλiAi+di1mi1+ci1+vλiAi+di2mi2+ci2+mi−mi∗−u−v.And,Γa,x=∫0xta−1e−tdt is an incomplete gamma function.Equations (62) and (63) cannot yield the explicit solutions for α^i0BL and α^i0BU. Thus, we can only obtain the numerical solution. Similarly, we can get α^ikBL,α^ikBU,k=1and2.Whenmi−mi∗=0, the parameter αik,k=0,1,and2, follows the gamma distribution with parameters mik+cik and λiAi+dik, and it is denoted by αik∼Gammamik+cik,λiAi+dik,k=0,1,and2. The 1001−γ% two-side BCI of αik,k=0,1,and2, can be acquired by the following equations:(65)γ2=∫0α^ikBUλiAi+dikmik+cikΓmik+cikαikmik+cik−1exp−αikλiAi+dikdαik,1−γ2=∫0α^ikBUλiAi+dikmik+cikΓmik+cikαikmik+cik−1exp−αikλiAi+dikdαik.In this case, ifαik,k=0,1,and2, is an integer, the 1001−γ% BCI is(66)χ1−γ/222mik+cik2λiAi+dik,χγ/222mik+cik2λiAi+dik.The highest posterior density (HPD) CIs ofαik,k=0,1,and2, say α^ikHL,α^ikHU, is the shortest interval and can be obtained from α^ikHL−α^ikHU=minα^ikBL−α^ikBU,k=0,1,and2.


## 5. Numerical Simulation

### 5.1. Simulation Study

Consider a two-level CSALT under AT-II PHCS. The subjected units have two dependent failure causes where the lifetime of the two failure causes are assumed to follow a MOBG distribution. Suppose the temperature is the accelerate stress, the accelerated stress levels areS1=50°C=323K and S2=80°C=353K, and the normal stress level is S0=25°C=298K.Under the stress levelSi,i=1and2, the sample size is n1=n2=30,40,and50, and the predetermined failure numbers are m1=10,15,20,25 and m2=15,20,25,30. The life tests are allowed to run over the time τ=0.5,1. It is assumed that the proportion of failure data which has unknown failure cause is 10%. Three different progressive censoring schemes are used in the CSALT, which are denoted as follows.Scheme 1:R1=R2=⋯=Rm−1=0andRm=n−mScheme 2:R1=n−mandR2=⋯=Rm−1=Rm=0Scheme 3:R1=R2=⋯=Rm/2=1,Rm+1/2=⋯=Rm−1=0andRm=n−m−m/2The parametersλi,αi0,αi1, and αi2 are given by the accelerated life functions as follows:(67)logλi=−6+1500/si,logai0=−3+800/si,logai1=−4+1000/si,logai2=−4+1200/si.Then, the initial values of the parameters areλ1,α10,α11,α12=0.2577,0.5926,0.4049,0.7521 and λ2,α20,α21,α22=0.1733,0.4801,0.3113,0.5484. To compute the BEs and HPD CIs of the parameters, the hyperparameters of the priors are given by ci0,ci1,ci2=1,1,3 and di0,di1,di2=1,1,1. The prior distribution of λi,i=1and2, is given in Table 1.Table 1
The prior distribution of the parameterλi,i=1and2.
λ10.20.3λ20.150.2Pλ1=λ1r1/21/2Pλ2=λ2r1/21/2Under different progressive censoring schemes, the process is repeated 1000 times, and then, the expected values of MLEs and BEs and mean squared errors (MSEs) of the unknown parameters are computed, and the results are shown in Tables2 and 3. The average interval lengths (AIL) and coverage percentages (CPs) of ACIs and HPD CIs with the given significant level (0.05) are also computed. The results are shown in Tables 4 and 5.Table 2
The MLEs and MSEs for the unknown parameters (τ=0.5).
n,m1,m2SchemeEstimator (MSE)λ1α10α11α12λ2α20α21α22(30,10,15)1MLE0.3702 (0.0998)0.7014 (0.1089)0.5158 (0.1067)0.8842 (0.1274)0.2689 (0.0873)0.5969 (0.1081)0.4376 (0.0967)0.6534 (0.1093)Bayes0.3498 (0.0884)0.6912 (0.0987)0.5036 (0.0957)0.8775 (0.1054)0.2517 (0.0782)0.5784 (0.0975)0.4128 (0.0897)0.6232 (0.0972)2MLE0.3668 (0.0987)0.7012 (0.1047)0.5178 (0.1017)0.8834 (0.1276)0.2687 (0.0879)0.5975 (0.1077)0.4377 (0.0999)0.6513 (0.1099)Bayes0.3484 (0.0825)0.6841 (0.0989)0.4878 (0.0935)0.6279 (0.1053)0.2512 (0.0785)0.5765 (0.0978)0.4086 (0.0899)0.6235 (0.0974)3MLE0.3677 (0.1017)0.7005 (0.1089)0.5175 (0.1019)0.8832 (0.1259)0.2669 (0.0957)0.5999 (0.1076)0.4372 (0.0972)0.6511 (0.1075)Bayes0.3489 (0.0822)0.6817 (0.0988)0.4976 (0.0932)0.8536 (0.1061)0.2511 (0.0791)0.5766 (0.0977)0.4095 (0.0895)0.6231 (0.0979)(30,15,20)1MLE0.3515 (0.0879)0.6895 (0.0992)0.4996 (0.0972)0.6333 (0.1162)0.2574 (0.0783)0.5873 (0.0989)0.4232 (0.0891)0.4513 (0.0978)Bayes0.3378 (0.0795)0.6789 (0.0907)0.4879 (0.0889)0.8489 (0.0992)0.2486 (0.0699)0.5703 (0.0838)0.4036 (0.0817)0.6177 (0.0897)2MLE0.3484 (0.0875)0.6884 (0.0992)0.5018 (0.0977)0.8699 (0.1164)0.2543 (0.0791)0.5884 (0.0985)0.4259 (0.0888)0.6417 (0.0981)Bayes0.3355 (0.0784)0.6747 (0.0906)0.4857 (0.0888)0.8481 (0.0987)0.2453 (0.0697)0.5654 (0.0832)0.4003 (0.0818)0.6158 (0.0894)3MLE0.3477 (0.0878)0.6876 (0.0998)0.5024 (0.0975)0.8696 (0.1164)0.2565 (0.0787)0.5886 (0.0997)0.4257 (0.0895)0.6413 (0.0982)Bayes0.3355 (0.0794)0.6725 (0.0905)0.4787 (0.0884)0.8452 (0.0987)0.2457 (0.0695)0.5655 (0.0844)0.3988 (0.0819)0.6139 (0.0893)(40,15,20)1MLE0.3379 (0.0816)0.6712 (0.0939)0.4916 (0.0871)0.8559 (0.1027)0.2416 (0.0725)0.5753 (0.0894)0.4118 (0.0829)0.6313 (0.0882)Bayes0.3255 (0.0711)0.6689 (0.0847)0.4685 (0.0768)0.8354 (0.0915)0.2398 (0.0604)0.5596 (0.0775)0.3975 (0.0753)0.6166 (0.0811)2MLE0.337 (0.0808)0.6775 (0.0936)0.4886 (0.0883)0.8594 (0.1031)0.2487 (0.0727)0.5764 (0.0881)0.4119 (0.0831)0.6327 (0.0871)Bayes0.3273 (0.0716)0.6672 (0.0852)0.4694 (0.0769)0.8376 (0.0917)0.2369 (0.0609)0.5581 (0.0772)0.3876 (0.0772)0.6165 (0.0817)3MLE0.3378 (0.0816)0.6786 (0.0932)0.4878 (0.0894)0.8593 (0.1029)0.2468 (0.0728)0.5763 (0.0895)0.4101 (0.0821)0.6368 (0.0897)Bayes0.3288 (0.0714)0.6691 (0.0859)0.4691 (0.0762)0.8378 (0.0919)0.2371 (0.0612)0.5568 (0.0778)0.3893 (0.0768)0.6045 (0.0813)(40,20,25)1MLE0.325 (0.0764)0.6611 (0.0849)0.4768 (0.0779)0.8438 (0.0945)0.2359 (0.0679)0.5667 (0.0812)0.3964 (0.0785)0.6269 (0.0811)Bayes0.3143 (0.0656)0.6571 (0.0776)0.4623 (0.0682)0.8294 (0.0897)0.2252 (0.0587)0.5508 (0.0705)0.3758 (0.0711)0.6044 (0.0732)2MLE0.3255 (0.0762)0.669 (0.0843)0.4758 (0.0771)0.8493 (0.0951)0.2375 (0.0682)0.5674 (0.0815)0.4044 (0.0794)0.6277 (0.0809)Bayes0.3155 (0.0657)0.6549 (0.0779)0.4589 (0.0687)0.8283 (0.0892)0.2275 (0.0583)0.5496 (0.0702)0.3779 (0.0713)0.6036 (0.0735)3MLE0.3235 (0.0764)0.6673 (0.0858)0.4686 (0.0773)0.8392 (0.0953)0.2388 (0.0681)0.5647 (0.0811)0.3988 (0.0789)0.6249 (0.0809)Bayes0.3153 (0.0653)0.6585 (0.0778)0.4578 (0.0684)0.8254 (0.0893)0.2369 (0.0585)0.5486 (0.0708)0.3764 (0.0712)0.5984 (0.0736)(50,20,25)1MLE0.3053 (0.0635)0.6538 (0.0758)0.4655 (0.0661)0.8281 (0.0863)0.2268 (0.0584)0.5466 (0.0718)0.3861 (0.0681)0.6129 (0.0711)Bayes0.2944 (0.0558)0.6406 (0.0675)0.4438 (0.0578)0.8192 (0.0783)0.2181 (0.0511)0.5324 (0.0625)0.3666 (0.0609)0.5893 (0.0633)2MLE0.3049 (0.0631)0.6596 (0.0757)0.4739 (0.0663)0.8295 (0.0859)0.2254 (0.0587)0.5542 (0.0714)0.3913 (0.0682)0.6128 (0.0717)Bayes0.2963 (0.0554)0.6393 (0.0681)0.4475 (0.0576)0.8186 (0.0787)0.2188 (0.0513)0.5337 (0.0626)0.3651 (0.0607)0.585 (0.0631)3MLE0.3099 (0.0633)0.6545 (0.0754)0.4669 (0.0664)0.8288 (0.0862)0.2283 (0.0586)0.5546 (0.0716)0.3889 (0.0683)0.6137 (0.0719)Bayes0.2989 (0.0552)0.6379 (0.0678)0.4461 (0.0575)0.8187 (0.0785)0.2185 (0.059)0.5348 (0.0624)0.3666 (0.0608)0.5881 (0.0632)(50,25,30)1MLE0.2892 (0.0512)0.6438 (0.0642)0.4551 (0.0518)0.8189 (0.0778)0.2164 (0.0487)0.5377 (0.0614)0.3747 (0.0528)0.6085 (0.0684)Bayes0.2876 (0.0419)0.6368 (0.0565)0.4329 (0.0509)0.8029 (0.0695)0.2089 (0.0409)0.5248 (0.0512)0.3586 (0.0482)0.5784 (0.0537)2MLE0.2902 (0.0511)0.6437 (0.0639)0.4548 (0.0517)0.8173 (0.0776)0.2174 (0.0485)0.5393 (0.0618)0.3773 (0.0527)0.6072 (0.0681)Bayes0.2851 (0.0411)0.6369 (0.0567)0.4356 (0.0502)0.8063 (0.0697)0.2074 (0.0408)0.5245 (0.0513)0.3592 (0.0485)0.5788 (0.0535)3MLE0.2874 (0.0513)0.6487 (0.0641)0.4548 (0.0514)0.8186 (0.0777)0.2187 (0.0488)0.5383 (0.0612)0.3773 (0.0525)0.6087 (0.0686)Bayes0.2736 (0.0418)0.6295 (0.0561)0.4373 (0.0508)0.8059 (0.0695)0.2077 (0.0411)0.5259 (0.0516)0.3603 (0.0484)0.5778 (0.0536)Table 3
The MLEs and MSEs for the unknown parameters (τ=1).
n,m1,m2SchemeEstimator (MSE)λ1α10α11α12λ2α20α21α22(30,10,15)1MLE0.3686 (0.0954)0.7009 (0.1027)0.5129 (0.1015)0.8813 (0.1257)0.2615 (0.0864)0.5946 (0.1074)0.4346 (0.0953)0.6516 (0.1078)Bayes0.3488 (0.0818)0.6897 (0.0985)0.4913 (0.0916)0.627 (0.1051)0.2472 (0.0749)0.5751 (0.0905)0.4053 (0.0895)0.6218 (0.0971)2MLE0.3617 (0.0946)0.7011 (0.1033)0.5124 (0.0977)0.8816 (0.1245)0.2646 (0.0872)0.5975 (0.1077)0.4334 (0.0973)0.6469 (0.1032)Bayes0.3472 (0.0812)0.6814 (0.0984)0.4923 (0.0917)0.6262 (0.1049)0.2496 (0.0775)0.5742 (0.9118)0.4054 (0.0898)0.6205 (0.0968)3MLE0.3662 (0.0996)0.6977 (0.1034)0.5117 (0.0966)0.8792 (0.1247)0.2668 (0.0899)0.5984 (0.1073)0.4346 (0.0959)0.6493 (0.1013)Bayes0.3439 (0.0811)0.6798 (0.0981)0.4933 (0.0914)0.8524 (0.1056)0.2493 (0.0765)0.5741 (0.0969)0.4051 (0.0899)0.6218 (0.0964)(30,15,20)1MLE0.3482 (0.0877)0.6854 (0.0982)0.4979 (0.0923)0.8698 (0.1155)0.2567 (0.0759)0.5854 (0.0981)0.4209 (0.0877)0.6385 (0.0971)Bayes0.3341 (0.0775)0.6774 (0.0902)0.483 (0.0876)0.844 (0.0989)0.2457 (0.0664)0.5695 (0.0821)0.3962 (0.0812)0.6149 (0.0886)2MLE0.3476 (0.0873)0.6871 (0.984)0.4939 (0.0926)0.8679 (0.1157)0.2532 (0.0782)0.5845 (0.0981)0.4206 (0.0876)0.6395 (0.0978)Bayes0.3317 (0.0772)0.6715 (0.0903)0.4819 (0.0875)0.8475 (0.0985)0.2439 (0.0667)0.5642 (0.0822)0.3959 (0.0819)0.6139 (0.0883)3MLE0.3466 (0.0875)0.6853 (0.0989)0.4958 (0.0924)0.8688 (0.1163)0.2549 (0.0784)0.5835 (0.0987)0.4208 (0.0874)0.6393 (0.0973)Bayes0.3325 (0.0778)0.6703 (0.0901)0.4769 (0.0867)0.8395 (0.0987)0.2427 (0.0667)0.5624 (0.0821)0.3955 (0.0818)0.6129 (0.0886)(40,15,20)1MLE0.3355 (0.0812)0.679 (0.0924)0.4864 (0.0856)0.8526 (0.1021)0.2508 (0.0719)0.5712 (0.0887)0.4083 (0.0815)0.6296 (0.0876)Bayes0.3255 (0.0702)0.6671 (0.0833)0.4681 (0.0756)0.8358 (0.0914)0.2417 (0.0603)0.5546 (0.0767)0.386 (0.0744)0.6025 (0.0807)2MLE0.3358 (0.0814)0.6755 (0.0914)0.4822 (0.0851)0.8588 (0.1026)0.2445 (0.0718)0.573 (0.0878)0.4077 (0.0814)0.6305 (0.0881)Bayes0.3254 (0.0706)0.6644 (0.0839)0.4669 (0.0756)0.8354 (0.0914)0.2312 (0.0605)0.5539 (0.0766)0.3856 (0.0748)0.6022 (0.0809)3MLE0.3368 (0.0818)0.6768 (0.0915)0.4865 (0.0855)0.8577 (0.1027)0.2448 (0.0713)0.5721 (0.0873)0.4091 (0.0812)0.6317 (0.0891)Bayes0.3256 (0.0708)0.668 (0.0837)0.4686 (0.0758)0.833 (0.0911)0.2348 (0.0607)0.5537 (0.0763)0.3879 (0.0748)0.6027 (0.0811)(40,20,25)1MLE0.3229 (0.0748)0.6655 (0.0836)0.4721 (0.0768)0.6633 (0.0941)0.246 (0.0661)0.4029 (0.0803)0.2338 (0.0777)0.471 (0.0802)Bayes0.3123 (0.0644)0.6523 (0.0765)0.4579 (0.0675)0.8257 (0.0885)0.2335 (0.0579)0.5483 (0.0696)0.3727 (0.0705)0.6018 (0.0731)2MLE0.3213 (0.0746)0.6656 (0.0833)0.473 (0.0766)0.8479 (0.0949)0.2365 (0.0669)0.5621 (0.0801)0.3973 (0.0772)0.6262 (0.0808)Bayes0.3127 (0.0642)0.6502 (0.0762)0.456 (0.0674)0.8259 (0.0888)0.2247 (0.0577)0.5489 (0.0697)0.3737 (0.0701)0.6019 (0.0729)3MLE0.3215 (0.0746)0.6625 (0.0839)0.4767 (0.0767)0.6692 (0.0952)0.2361 (0.0662)0.5628 (0.0809)0.2324 (0.0774)0.6212 (0.0801)Bayes0.3141 (0.0641)0.651 (0.0761)0.4563 (0.0679)0.8232 (0.0887)0.2241 (0.0573)0.5479 (0.0699)0.3727 (0.0703)0.5931 (0.0731)(50,20,25)1MLE0.3016 (0.0621)0.6596 (0.0751)0.4641 (0.0656)0.8265 (0.0852)0.2343 (0.0576)0.5522 (0.0709)0.3852 (0.0675)0.6114 (0.0707)Bayes0.2924 (0.0549)0.6395 (0.0674)0.4429 (0.0567)0.8187 (0.0779)0.2271 (0.0509)0.5319 (0.0623)0.364 (0.0605)0.5807 (0.0629)2MLE0.3017 (0.0627)0.6587 (0.0748)0.3418 (0.0657)0.8281 (0.0851)0.2248 (0.0582)0.5515 (0.0706)0.3879 (0.0674)0.6114 (0.0711)Bayes0.2916 (0.0549)0.6387 (0.0679)0.4457 (0.0562)0.8152 (0.0778)0.2174 (0.0505)0.5329 (0.0619)0.3644 (0.0607)0.5832 (0.0621)3MLE0.3093 (0.0626)0.6537 (0.0749)0.4651 (0.0655)0.8274 (0.0859)0.2269 (0.0577)0.5518 (0.0701)0.3852 (0.0679)0.6103 (0.0712)Bayes0.297 (0.0544)0.6359 (0.0677)0.4455 (0.0564)0.8176 (0.0781)0.2169 (0.0507)0.5325 (0.0621)0.3628 (0.0603)0.5846 (0.0624)(50,25,30)1MLE0.2828 (0.0508)0.6414 (0.0639)0.4527 (0.0511)0.8164 (0.0761)0.2251 (0.0471)0.5362 (0.0607)0.3724 (0.0511)0.6059 (0.0681)Bayes0.2752 (0.0407)0.6345 (0.0563)0.4318 (0.0507)0.8033 (0.0688)0.2169 (0.0404)0.4379 (0.0508)0.3552 (0.0479)0.5742 (0.0524)2MLE0.2892 (0.0509)0.6423 (0.0637)0.4517 (0.0507)0.8151 (0.0768)0.2163 (0.0478)0.5391 (0.0611)0.3747 (0.0518)0.4871 (0.0678)Bayes0.273 (0.0403)0.6233 (0.0561)0.4312 (0.0509)0.8016 (0.0685)0.2046 (0.0405)0.5216 (0.0509)0.3585 (0.0482)0.5762 (0.0531)3MLE0.2851 (0.0509)0.6443 (0.0635)0.4508 (0.0505)0.815 (0.0769)0.2152 (0.0477)0.5377 (0.0609)0.3755 (0.0517)0.6072 (0.0682)Bayes0.2716 (0.0406)0.6276 (0.0558)0.4352 (0.0506)0.803 (0.0687)0.2062 (0.0407)0.5224 (0.0507)0.3595 (0.0481)0.5766 (0.0527)Table 4
The CPs and AILs of the confidence intervals for unknown parameters (τ=0.5).
n,m1,m2SchemeCP (AIL)λ1α10α11α12λ2α20α21α22(30,10,15)1MLE84.7 (0.2892)84.7 (0.3312)84.4 (0.2763)85.1 (0.3762)85.2 (0.2195)84.6 (0.2953)84.7 (0.2682)85.8 (0.3152)Bayes88.1 (0.2768)87.1 (0.3198)88.1 (0.2684)88.7 (0.3638)89.1 (0.2079)90.2 (0.2861)88.4 (0.2586)89.3 (0.3055)2MLE84.3 (0.2868)84.5 (0.3237)85.9 (0.2748)85.2 (0.3724)85.1 (0.2174)84.2 (0.2972)84.5 (0.2684)85.3 (0.3154)Bayes87.9 (0.2732)88.2 (0.3158)89.2 (0.2681)88.7 (0.3628)89.5 (0.2738)90.1 (0.2822)88.2 (0.2558)89.1 (0.3037)3MLE84.2 (0.2872)84.1 (0.3254)85.1 (0.2775)85 (0.3713)85.2 (0.2174)84.9 (0.2961)84.4 (0.2653)85.3 (0.3171)Bayes87.2 (0.2752)88.1 (0.3149)89.1 (0.2669)88.1 (0.3652)89.4 (0.2044)89.9 (0.2846)88.2 (0.2393)89.2 (0.3024)(30,15,20)1MLE86 (0.2675)86.3 (0.3084)85.2 (0.2569)86.5 (0.3546)87 (0.1989)85.2 (0.2773)86.8 (0.2489)87 (0.2965)Bayes90.3 (0.2546)89.2 (0.2969)89.6 (0.2478)89.2 (0.3412)90.3 (0.1891)91.1 (0.2684)89.9 (0.2372)91.1 (0.2858)2MLE85.6 (0.2672)85.7 (0.3058)86.3 (0.2545)87.2 (0.3553)86.7 (0.1987)86.2 (0.2785)85.8 (0.2487)87.6 (0.2966)Bayes89.6 (0.2549)90.3 (0.2964)89.4 (0.2469)89.3 (0.3454)90.2 (0.1871)91.8 (0.2684)89.2 (0.2379)90.2 (0.2845)3MLE85.4 (0.2689)85.6 (0.3074)86.4 (0.2582)86 (0.3591)87 (0.2025)86.2 (0.2778)85.6 (0.2483)86.8 (0.2954)Bayes89.1 (0.2571)89.7 (0.2956)90.1 (0.2448)89.3 (0.3447)90.4 (0.1887)90.1 (0.2676)89.3 (0.2368)90.2 (0.2822)(40,15,20)1MLE87.4 (0.2527)87.7 (0.2873)86.1 (0.2366)86.8 (0.3362)88 (0.1831)87.3 (0.2569)88 (0.2287)89 (0.2778)Bayes91.4 (0.2376)92.1 (0.2765)92.9 (0.2264)91.1 (0.3254)91.6 (0.1683)92.6 (0.2488)92 (0.2185)92.3 (0.2687)2MLE87 (0.2545)86.6 (0.2853)88.4 (0.2347)88.5 (0.3376)88.3 (0.1826)88.8 (0.2595)87.4 (0.2273)89.4 (0.2795)Bayes91.5 (0.2423)92.3 (0.2733)91.6 (0.2276)91.6 (0.3262)91.6 (0.1667)92.2 (0.2457)91.5 (0.2177)92.8 (0.2693)3MLE87.1 (0.2579)86.9 (0.2893)87.9 (0.2376)88.1 (0.3368)88 (0.1852)88.1 (0.2548)87.2 (0.2297)88.7 (0.2762)Bayes91.1 (0.2425)92 (0.2758)91.2 (0.2269)91.1 (0.3245)91.5 (0.1685)92.2 (0.2481)91.3 (0.2183)92.3 (0.2679)(40,20,25)1MLE89 (0.2333)89 (0.2736)88 (0.2149)88.3 (0.3258)90 (0.1684)89.6 (0.2335)90 (0.2142)90.2 (0.2611)Bayes94.2 (0.2273)94.1 (0.2543)94.5 (0.2076)93.5 (0.3062)94.1 (0.1482)94.2 (0.2246)93.3 (0.1964)93.2 (0.2563)2MLE89.5 (0.2362)89 (0.2698)89.5 (0.2182)89.5 (0.3221)89.2 (0.1651)89.8 (0.2361)89 (0.2138)91 (0.2578)Bayes94.4 (0.2256)94.9 (0.2577)94.8 (0.2086)94.3 (0.3076)93.9 (0.1463)94.8 (0.2265)93.5 (0.1967)94.4 (0.2467)3MLE89.3 (0.2344)88.2 (0.2756)89.2 (0.2226)89.3 (0.3221)89.7 (0.1692)89.7 (0.2368)88.7 (0.2139)89.9 (0.2572)Bayes94.3 (0.2272)94.8 (0.2566)94 (0.2077)94.3 (0.3089)93.3 (0.1469)94.1 (0.2344)93.3 (0.1987)94.1 (0.2452)(50,20,25)1MLE90.9 (0.2166)89.9 (0.2547)89.9 (0.2051)89.5 (0.3089)90.8 (0.1474)90.8 (0.2148)91.8 (0.1989)91.3 (0.248)Bayes95.7 (0.2066)96.1 (0.2342)95.6 (0.1878)95 (0.2854)96.8 (0.1324)95.9 (0.2078)95.2 (0.1732)95.1 (0.2283)2MLE90.8 (0.2185)89.2 (0.2532)90.7 (0.2025)90.2 (0.3097)90.9 (0.1442)90.1 (0.2176)90.8 (0.1942)91.7 (0.2423)Bayes95.2 (0.2068)96.2 (0.2338)95.8 (0.1853)97.8 (0.2887)96.2 (0.1292)96.7 (0.2083)97.8 (0.1784)96.4 (0.2256)3MLE90.5 (0.2169)89.1 (0.2541)90.2 (0.2015)90.1 (0.3076)90.3 (0.1458)90.1 (0.2234)90.1 (0.1935)91.1 (0.2373)Bayes95.1 (0.2089)96.1 (0.2429)95.2 (0.1871)97 (0.2895)96.4 (0.1282)96.3 (0.2143)96.8 (0.1763)96.3 (0.2245)(50,25,30)1MLE92.7 (0.1967)91.7 (0.2357)91.6 (0.1879)92.1 (0.2865)91.5 (0.1263)91.6 (0.1971)92.4 (0.1784)93 (0.2266)Bayes97.1 (0.1866)97.8 (0.2151)97.7 (0.1657)97.9 (0.2648)98.3 (0.1132)97.9 (0.1843)98.4 (0.1579)97.8 (0.2058)2MLE91.6 (0.1971)91.4 (0.2356)92.6 (0.1857)92.2 (0.2859)92.5 (0.1287)92.4 (0.1954)92.8 (0.1784)93.5 (0.2283)Bayes97.6 (0.1832)97.3 (0.2152)97.5 (0.1669)98 (0.2666)98.6 (0.1187)97.5 (0.1824)98.7 (0.1576)97.8 (0.2073)3MLE91.4 (0.1978)91.2 (0.2352)92.1 (0.1834)92.2 (0.2858)92.1 (0.1266)92.2 (0.2026)92.1 (0.1728)93.2 (0.2134)Bayes97.5 (0.1852)97.1 (0.2276)97.9 (0.1653)98.1 (0.2673)97.5 (0.1142)97.1 (0.1945)98 (0.1575)97.8 (0.2065)Table 5
The CPs and AILs of the confidence intervals for unknown parameters (τ=1).
n,m1,m2SchemeCP (AIL)λ1α10α11α12λ2α20α21α22(30,10,15)1MLE84.7 (0.2886)84.3 (0.3294)84.3 (0.2739)84.9 (0.3737)85.2 (0.2186)84.5 (0.2922)84.3 (0.2662)85.1 (0.3135)Bayes88.1 (0.2737)87.3 (0.3177)87.9 (0.2675)88.4 (0.3619)88.9 (0.2056)90.2 (0.2832)88.4 (0.2563)89.1 (0.3027)2MLE84.5 (0.2875)84.2 (0.3246)84.3 (0.2723)84.6 (0.3732)85.1 (0.2168)84.2 (0.2952)84.1 (0.269)84.8 (0.3172)Bayes88.1 (0.2755)87.1 (0.3168)87.6 (0.2659)88.2 (0.3625)88.2 (0.2027)89.7 (0.2821)88.2 (0.2528)89.2 (0.3018)3MLE84.3 (0.2881)84.2 (0.3239)84.2 (0.2751)84.4 (0.3702)85 (0.2189)84.1 (0.297)84.2 (0.2611)84.6 (0.3138)Bayes88.1 (0.2748)87.3 (0.3127)87.4 (0.2666)88.1 (0.3622)88.1 (0.1989)89.5 (0.2812)88.3 (0.2532)89.1 (0.3015)(30,15,20)1MLE85.7 (0.2694)86.2 (0.3076)85.8 (0.2577)86.2 (0.3544)86.5 (0.1999)86.2 (0.2785)85.3 (0.2478)86.9 (0.2948)Bayes90.5 (0.2569)89.1 (0.2965)89.6 (0.2485)89.1 (0.3426)90.1 (0.1885)90.1 (0.2676)89.4 (0.2388)90.1 (0.2869)2MLE85.5 (0.2684)86.1 (0.3065)85.7 (0.2588)86.4 (0.3589)86.4 (0.1997)86.1 (0.2797)86 (0.2481)86.2 (0.2987)Bayes90.2 (0.2596)89.3 (0.2974)89.6 (0.2486)89 (0.3456)90 (0.1864)90.1 (0.2679)89.3 (0.2356)90.1 (0.2852)3MLE85.4 (0.2702)86.1 (0.3096)85.7 (0.2571)86.2 (0.3605)89.3 (0.2041)86.2 (0.2798)85.8 (0.2495)86.2 (0.2962)Bayes90 (0.2583)89.2 (0.2977)89.3 (0.2463)88.8 (0.3414)89.9 (0.1887)90.1 (0.2675)89.2 (0.2369)90.2 (0.283)(40,15,20)1MLE87.2 (0.2513)87.3 (0.2888)86.7 (0.2369)86.7 (0.3377)87.9 (0.1812)87.3 (0.2589)87.6 (0.2277)88.5 (0.2765)Bayes91.2 (0.2426)91.6 (0.2713)92.6 (0.2296)91 (0.3249)91.4 (0.1697)92.4 (0.2484)91.9 (0.2197)92.2 (0.2696)2MLE87.2 (0.2537)87.2 (0.2876)86.3 (0.2318)87.8 (0.3399)87.8 (0.1807)87.2 (0.2596)87.5 (0.2298)88.2 (0.2787)Bayes91.1 (0.2419)91.2 (0.2737)92.7 (0.2278)91 (0.3232)91.2 (0.1687)92.4 (0.2437)91.7 (0.2189)92 (0.2683)3MLE87 (0.2562)87.1 (0.2908)86.1 (0.2396)87.7 (0.3356)87.8 (0.1874)87.2 (0.2585)87.5 (0.2303)88.2 (0.2785)Bayes90.9 (0.2412)91.2 (0.2765)92.1 (0.2257)90.8 (0.3275)91.2 (0.1667)92.2 (0.2493)91.7 (0.2141)92 (0.2625)(40,20,25)1MLE88.6 (0.2346)88.9 (0.2724)87.9 (0.2184)88.2 (0.3212)89.5 (0.1686)89.4 (0.2342)89.1 (0.2139)90.1 (0.2601)Bayes93.8 (0.2265)94 (0.2584)94.3 (0.2093)93.2 (0.3086)94.1 (0.1499)94.1 (0.2253)93.7 (0.1997)94 (0.2579)2MLE88.4 (0.2369)88.7 (0.2692)87.9 (0.2193)88.3 (0.3215)89.2 (0.1677)89.3 (0.2372)89.1 (0.2104)90 (0.2597)Bayes93.8 (0.2273)93.9 (0.2595)94.4 (0.2091)93.1 (0.3093)93.9 (0.1476)94 (0.2295)93.4 (0.1999)93.8 (0.2448)3MLE88.4 (0.235)88.7 (0.2719)87.8 (0.2219)88 (0.3241)89.2 (0.1685)89.3 (0.2393)89 (0.2123)89.6 (0.2587)Bayes93.4 (0.2286)93.9 (0.2598)94.2 (0.2085)93.3 (0.3108)93.8 (0.1439)93.8 (0.2314)93.3 (0.1979)93.6 (0.2436)(50,20,25)1MLE90.5 (0.2164)89.6 (0.2561)89.6 (0.2048)90.1 (0.3099)90.3 (0.1474)90.5 (0.2186)91.4 (0.1964)91.3 (0.2486)Bayes95.4 (0.2072)96.1 (0.2337)95.6 (0.1853)95.1 (0.2876)96.3 (0.1314)95.4 (0.2089)95.1 (0.1776)95 (0.2243)2MLE90.3 (0.2195)89.5 (0.2517)89.5 (0.2015)90 (0.3093)90.1 (0.1481)90.1 (0.2192)91.4 (0.1947)91.2 (0.2401)Bayes95.4 (0.2095)95.8 (0.2359)95.3 (0.1877)95 (0.2892)96.2 (0.1287)95.3 (0.2107)95.1 (0.1795)95 (0.2261)3MLE90.2 (0.2167)89.5 (0.2538)89.4 (0.2029)89.5 (0.3085)89.7 (0.1497)90.1 (0.2213)91.3 (0.1942)91.2 (0.2389)Bayes94.3 (0.2104)95.3 (0.2413)95.2 (0.1895)95 (0.2886)96.1 (0.1274)95.2 (0.2171)95.1 (0.1778)94.9 (0.2216)(50,25,30)1MLE92.7 (0.1987)91.4 (0.2356)91.3 (0.1864)92.6 (0.2885)91.4 (0.1237)92.3 (0.1993)92.3 (0.1763)93 (0.224)Bayes97.1 (0.1857)97.3 (0.2134)97.2 (0.1646)97.2 (0.2661)98.2 (0.1114)97.5 (0.1856)98.4 (0.1575)97.8 (0.2037)2MLE92.6 (0.1972)91.4 (0.2364)91.1 (0.1876)92.4 (0.2869)91.4 (0.1261)92.2 (0.1927)92.2 (0.1769)92.8 (0.2222)Bayes97.1 (0.1829)97.3 (0.2155)97.2 (0.1655)97.1 (0.2658)98.2 (0.1142)97.4 (0.1887)98.3 (0.1586)97.8 (0.2062)3MLE92.6 (0.1957)91.4 (0.2357)91.1 (0.1835)92.4 (0.2886)91.4 (0.1274)92.2 (0.2013)92.2 (0.1733)92.8 (0.2162)Bayes97.8 (0.1848)97.3 (0.2258)97.2 (0.1642)97.1 (0.2679)98.2 (0.1145)97.4 (0.1923)98.3 (0.1566)97.8 (0.2045)From Tables2 and 3, it can be observed that the MSEs of the MLEs and BEs for the unknown parameters are decreased with the sample size n increase or the effective sample size m increases, and the MLEs and BEs of the unknown parameters are closer to the true values. For the bigger τ, the MSEs for the MLEs and Bayes of unknown parameters both are the smaller. In addition, the difference under three difference progressive censored schemes is not sensible. Furthermore, the BEs of the unknown parameters are better than MLEs, and its MSEs are always less than the MLEs.From Tables4 and 5, the CPs of ACIs and HPD CIs for the unknown parameters are increased, and meanwhile, the AILs of the ACIs and the HPD CIs are decreased with the sample size n or the effective sample size m increase. For the fixed n, m, and τ, the CPs of HPD CIs are more stable than the ACIs for the unknown parameters.


### 5.2. Real Data Analysis

To illustrate the proposed methods, a real data is selected from Ebrahimi [25], of which the accelerated stress is voltage, the stress levels are S1=10V,S2=15V,andS3=20V, and the normal stress level is S0=5V. Furthermore, for each stress level, the life tests are allowed to run over the time τ1,τ2,τ3=2,1.5,1, the numbers under the test were n1,n2,n3=15,15,15, the expected failure number are m1,m2,m3=11,12,13, and the progressive censoring scheme is R1=R2=⋯=Rmi−1=0andRmi=ni−mi,i=1,2,and3.Assume that the unknown failure cause of the failure data proportion is 20%, and then, the failure data and the corresponding failure causes under each stress level are presented in Table6.Table 6
The failure data with the dependent competing risks’ model.
S1(0.04, 0)(0.10, 1)(0.16, 0)(0.17,∗)(0.42, 2)(0.46, 0)(0.47,∗)(0.47, 1)(1.05, 1)(1.50, 2)(1.80, 0)S2(0.02, 1)(0.06, 2)(0.27,∗)(0.30, 2)(0.30, 2)(0.34, 1)(0.36,∗)(0.70, 2)(0.80, 2)(0.80, 0)(0.92,∗)(1.06, 2)S3(0.07, 2)(0.13, 1)(0.20,∗)(0.29, 1)(0.30, 2)(0.30,∗)(0.30, 1)(0.33, 2)(0.33,∗)(0.40, 0)(0.40, 1)(0.63,∗)(0.82, 0)For the Bayes analysis, we select small values of the hyperparametersci0=ci1=ci2=di0=di1=di2=0.0001, which gives noninformation priors [26]. We make the data in Table 6 be brought to the above mode and obtain the point estimation and interval estimation of the unknown parameters. The results are shown in Table 7.Table 7
Estimations and 95% confidence and incredible intervals.
EstimatorMLEACIBayesHPDλ10.3133(0.0911, 0.5982)0.3027(0.0953, 0.5976)α101.7241(0.9851, 2.4724)1.7163(0.9735, 2.3853)α110.8616(0.3953, 1.1285)0.8473(0.4075, 1.1153)α120.8247(0.4261, 1.1176)0.8193(0.4378, 1.1104)λ20.8359(0.3956, 1.1432)0.8287(0.3918, 1.1427)α201.4937(0.6853, 2.2589)1.4517(0.6978, 2.2667)α210.7745(0.4137, 1.0134)0.7689(0.4166, 1.0089)α221.1201(0.6176, 1.6792)1.0816(0.6186, 1.6811)λ31.1024(0.5983, 1.5925)1.0673(0.6017, 1.5937)α302.3691(1.4327, 3.2896)2.2164(1.4673, 3.2576)α311.4151(0.8221, 2.1074)1.4091(0.8226, 2.0974)α321.5121(0.8831, 2.2146)1. 4973(0.8875, 2.2121)From Table7, we can obtain that there is little difference between the MLEs and BEs of the unknown parameters. The lengths for the HPD CIs are shorter than the ACIs in most cases. For the given significance value, the accuracy of the HPD CIs is better than the ACIs.Here, aiming at the above data analysis, there exists a question on how good the proposed MOBG model fits the data. That is, there have been multiple goodness-of-fit testing methods for univariate distributions. However, for bivariate distribution, the goodness-of-fit testing methods are less. Thus, to verify the above fit results of data and distribution, we mainly used the method which is from the Kundu and Gupta [27], and we verified two margins and the minimum lifetime by using the Kolmogorov–Smirnov test (K–S test).Give the significance valueγ=5%, the K–S statistic and the corresponding p values under different failure numbers are obtained, and the result is presented in Table 8. The comparison charts for empirical distribution and its fitted distribution of the sample under each stress are shown in Figures 1–3.Table 8
The K–S statistics and the correspondingp value.
DistributionFt;λ1,α101Ft;λ1,α102Ft;λ1,α1Ft;λ2,α201Ft;λ2,α202Ft;λ2,α2Ft;λ3,α301Ft;λ3,α302Ft;λ3,α3K–S0.24430.14350.16890.15150.17150.20350.17370.20890.2388p0.55620.95360.86200.90760.81570.63250.76760.63240.5873Figure 1
Comparison chart for empirical distribution and its fitted distribution underS1 level. (a) The comparison chart when α=α101. (b) The comparison chart when α=α102. (c) The comparison chart when α=α1.
(a)(b)(c)Figure 2
Comparison chart for empirical distribution and its fitted distribution underS2 level. (a) The comparison chart when α=α201. (b) The comparison chart when α=α202. (c) The comparison chart when α=α2.
(a)(b)(c)Figure 3
Comparison chart for empirical distribution and its fitted distribution underS3 level. (a) The comparison chart when α=α301. (b) The comparison chart when α=α302. (c) The comparison chart when α=α3.
(a)(b)(c)From Table8, we can obtain that all the corresponding p values are bigger than the given significance value. It indicates that the failure data comes from the MOBG distribution.


## 5.1. Simulation Study

Consider a two-level CSALT under AT-II PHCS. The subjected units have two dependent failure causes where the lifetime of the two failure causes are assumed to follow a MOBG distribution. Suppose the temperature is the accelerate stress, the accelerated stress levels areS1=50°C=323K and S2=80°C=353K, and the normal stress level is S0=25°C=298K.Under the stress levelSi,i=1and2, the sample size is n1=n2=30,40,and50, and the predetermined failure numbers are m1=10,15,20,25 and m2=15,20,25,30. The life tests are allowed to run over the time τ=0.5,1. It is assumed that the proportion of failure data which has unknown failure cause is 10%. Three different progressive censoring schemes are used in the CSALT, which are denoted as follows.Scheme 1:R1=R2=⋯=Rm−1=0andRm=n−mScheme 2:R1=n−mandR2=⋯=Rm−1=Rm=0Scheme 3:R1=R2=⋯=Rm/2=1,Rm+1/2=⋯=Rm−1=0andRm=n−m−m/2The parametersλi,αi0,αi1, and αi2 are given by the accelerated life functions as follows:(67)logλi=−6+1500/si,logai0=−3+800/si,logai1=−4+1000/si,logai2=−4+1200/si.Then, the initial values of the parameters areλ1,α10,α11,α12=0.2577,0.5926,0.4049,0.7521 and λ2,α20,α21,α22=0.1733,0.4801,0.3113,0.5484. To compute the BEs and HPD CIs of the parameters, the hyperparameters of the priors are given by ci0,ci1,ci2=1,1,3 and di0,di1,di2=1,1,1. The prior distribution of λi,i=1and2, is given in Table 1.Table 1
The prior distribution of the parameterλi,i=1and2.
λ10.20.3λ20.150.2Pλ1=λ1r1/21/2Pλ2=λ2r1/21/2Under different progressive censoring schemes, the process is repeated 1000 times, and then, the expected values of MLEs and BEs and mean squared errors (MSEs) of the unknown parameters are computed, and the results are shown in Tables2 and 3. The average interval lengths (AIL) and coverage percentages (CPs) of ACIs and HPD CIs with the given significant level (0.05) are also computed. The results are shown in Tables 4 and 5.Table 2
The MLEs and MSEs for the unknown parameters (τ=0.5).
n,m1,m2SchemeEstimator (MSE)λ1α10α11α12λ2α20α21α22(30,10,15)1MLE0.3702 (0.0998)0.7014 (0.1089)0.5158 (0.1067)0.8842 (0.1274)0.2689 (0.0873)0.5969 (0.1081)0.4376 (0.0967)0.6534 (0.1093)Bayes0.3498 (0.0884)0.6912 (0.0987)0.5036 (0.0957)0.8775 (0.1054)0.2517 (0.0782)0.5784 (0.0975)0.4128 (0.0897)0.6232 (0.0972)2MLE0.3668 (0.0987)0.7012 (0.1047)0.5178 (0.1017)0.8834 (0.1276)0.2687 (0.0879)0.5975 (0.1077)0.4377 (0.0999)0.6513 (0.1099)Bayes0.3484 (0.0825)0.6841 (0.0989)0.4878 (0.0935)0.6279 (0.1053)0.2512 (0.0785)0.5765 (0.0978)0.4086 (0.0899)0.6235 (0.0974)3MLE0.3677 (0.1017)0.7005 (0.1089)0.5175 (0.1019)0.8832 (0.1259)0.2669 (0.0957)0.5999 (0.1076)0.4372 (0.0972)0.6511 (0.1075)Bayes0.3489 (0.0822)0.6817 (0.0988)0.4976 (0.0932)0.8536 (0.1061)0.2511 (0.0791)0.5766 (0.0977)0.4095 (0.0895)0.6231 (0.0979)(30,15,20)1MLE0.3515 (0.0879)0.6895 (0.0992)0.4996 (0.0972)0.6333 (0.1162)0.2574 (0.0783)0.5873 (0.0989)0.4232 (0.0891)0.4513 (0.0978)Bayes0.3378 (0.0795)0.6789 (0.0907)0.4879 (0.0889)0.8489 (0.0992)0.2486 (0.0699)0.5703 (0.0838)0.4036 (0.0817)0.6177 (0.0897)2MLE0.3484 (0.0875)0.6884 (0.0992)0.5018 (0.0977)0.8699 (0.1164)0.2543 (0.0791)0.5884 (0.0985)0.4259 (0.0888)0.6417 (0.0981)Bayes0.3355 (0.0784)0.6747 (0.0906)0.4857 (0.0888)0.8481 (0.0987)0.2453 (0.0697)0.5654 (0.0832)0.4003 (0.0818)0.6158 (0.0894)3MLE0.3477 (0.0878)0.6876 (0.0998)0.5024 (0.0975)0.8696 (0.1164)0.2565 (0.0787)0.5886 (0.0997)0.4257 (0.0895)0.6413 (0.0982)Bayes0.3355 (0.0794)0.6725 (0.0905)0.4787 (0.0884)0.8452 (0.0987)0.2457 (0.0695)0.5655 (0.0844)0.3988 (0.0819)0.6139 (0.0893)(40,15,20)1MLE0.3379 (0.0816)0.6712 (0.0939)0.4916 (0.0871)0.8559 (0.1027)0.2416 (0.0725)0.5753 (0.0894)0.4118 (0.0829)0.6313 (0.0882)Bayes0.3255 (0.0711)0.6689 (0.0847)0.4685 (0.0768)0.8354 (0.0915)0.2398 (0.0604)0.5596 (0.0775)0.3975 (0.0753)0.6166 (0.0811)2MLE0.337 (0.0808)0.6775 (0.0936)0.4886 (0.0883)0.8594 (0.1031)0.2487 (0.0727)0.5764 (0.0881)0.4119 (0.0831)0.6327 (0.0871)Bayes0.3273 (0.0716)0.6672 (0.0852)0.4694 (0.0769)0.8376 (0.0917)0.2369 (0.0609)0.5581 (0.0772)0.3876 (0.0772)0.6165 (0.0817)3MLE0.3378 (0.0816)0.6786 (0.0932)0.4878 (0.0894)0.8593 (0.1029)0.2468 (0.0728)0.5763 (0.0895)0.4101 (0.0821)0.6368 (0.0897)Bayes0.3288 (0.0714)0.6691 (0.0859)0.4691 (0.0762)0.8378 (0.0919)0.2371 (0.0612)0.5568 (0.0778)0.3893 (0.0768)0.6045 (0.0813)(40,20,25)1MLE0.325 (0.0764)0.6611 (0.0849)0.4768 (0.0779)0.8438 (0.0945)0.2359 (0.0679)0.5667 (0.0812)0.3964 (0.0785)0.6269 (0.0811)Bayes0.3143 (0.0656)0.6571 (0.0776)0.4623 (0.0682)0.8294 (0.0897)0.2252 (0.0587)0.5508 (0.0705)0.3758 (0.0711)0.6044 (0.0732)2MLE0.3255 (0.0762)0.669 (0.0843)0.4758 (0.0771)0.8493 (0.0951)0.2375 (0.0682)0.5674 (0.0815)0.4044 (0.0794)0.6277 (0.0809)Bayes0.3155 (0.0657)0.6549 (0.0779)0.4589 (0.0687)0.8283 (0.0892)0.2275 (0.0583)0.5496 (0.0702)0.3779 (0.0713)0.6036 (0.0735)3MLE0.3235 (0.0764)0.6673 (0.0858)0.4686 (0.0773)0.8392 (0.0953)0.2388 (0.0681)0.5647 (0.0811)0.3988 (0.0789)0.6249 (0.0809)Bayes0.3153 (0.0653)0.6585 (0.0778)0.4578 (0.0684)0.8254 (0.0893)0.2369 (0.0585)0.5486 (0.0708)0.3764 (0.0712)0.5984 (0.0736)(50,20,25)1MLE0.3053 (0.0635)0.6538 (0.0758)0.4655 (0.0661)0.8281 (0.0863)0.2268 (0.0584)0.5466 (0.0718)0.3861 (0.0681)0.6129 (0.0711)Bayes0.2944 (0.0558)0.6406 (0.0675)0.4438 (0.0578)0.8192 (0.0783)0.2181 (0.0511)0.5324 (0.0625)0.3666 (0.0609)0.5893 (0.0633)2MLE0.3049 (0.0631)0.6596 (0.0757)0.4739 (0.0663)0.8295 (0.0859)0.2254 (0.0587)0.5542 (0.0714)0.3913 (0.0682)0.6128 (0.0717)Bayes0.2963 (0.0554)0.6393 (0.0681)0.4475 (0.0576)0.8186 (0.0787)0.2188 (0.0513)0.5337 (0.0626)0.3651 (0.0607)0.585 (0.0631)3MLE0.3099 (0.0633)0.6545 (0.0754)0.4669 (0.0664)0.8288 (0.0862)0.2283 (0.0586)0.5546 (0.0716)0.3889 (0.0683)0.6137 (0.0719)Bayes0.2989 (0.0552)0.6379 (0.0678)0.4461 (0.0575)0.8187 (0.0785)0.2185 (0.059)0.5348 (0.0624)0.3666 (0.0608)0.5881 (0.0632)(50,25,30)1MLE0.2892 (0.0512)0.6438 (0.0642)0.4551 (0.0518)0.8189 (0.0778)0.2164 (0.0487)0.5377 (0.0614)0.3747 (0.0528)0.6085 (0.0684)Bayes0.2876 (0.0419)0.6368 (0.0565)0.4329 (0.0509)0.8029 (0.0695)0.2089 (0.0409)0.5248 (0.0512)0.3586 (0.0482)0.5784 (0.0537)2MLE0.2902 (0.0511)0.6437 (0.0639)0.4548 (0.0517)0.8173 (0.0776)0.2174 (0.0485)0.5393 (0.0618)0.3773 (0.0527)0.6072 (0.0681)Bayes0.2851 (0.0411)0.6369 (0.0567)0.4356 (0.0502)0.8063 (0.0697)0.2074 (0.0408)0.5245 (0.0513)0.3592 (0.0485)0.5788 (0.0535)3MLE0.2874 (0.0513)0.6487 (0.0641)0.4548 (0.0514)0.8186 (0.0777)0.2187 (0.0488)0.5383 (0.0612)0.3773 (0.0525)0.6087 (0.0686)Bayes0.2736 (0.0418)0.6295 (0.0561)0.4373 (0.0508)0.8059 (0.0695)0.2077 (0.0411)0.5259 (0.0516)0.3603 (0.0484)0.5778 (0.0536)Table 3
The MLEs and MSEs for the unknown parameters (τ=1).
n,m1,m2SchemeEstimator (MSE)λ1α10α11α12λ2α20α21α22(30,10,15)1MLE0.3686 (0.0954)0.7009 (0.1027)0.5129 (0.1015)0.8813 (0.1257)0.2615 (0.0864)0.5946 (0.1074)0.4346 (0.0953)0.6516 (0.1078)Bayes0.3488 (0.0818)0.6897 (0.0985)0.4913 (0.0916)0.627 (0.1051)0.2472 (0.0749)0.5751 (0.0905)0.4053 (0.0895)0.6218 (0.0971)2MLE0.3617 (0.0946)0.7011 (0.1033)0.5124 (0.0977)0.8816 (0.1245)0.2646 (0.0872)0.5975 (0.1077)0.4334 (0.0973)0.6469 (0.1032)Bayes0.3472 (0.0812)0.6814 (0.0984)0.4923 (0.0917)0.6262 (0.1049)0.2496 (0.0775)0.5742 (0.9118)0.4054 (0.0898)0.6205 (0.0968)3MLE0.3662 (0.0996)0.6977 (0.1034)0.5117 (0.0966)0.8792 (0.1247)0.2668 (0.0899)0.5984 (0.1073)0.4346 (0.0959)0.6493 (0.1013)Bayes0.3439 (0.0811)0.6798 (0.0981)0.4933 (0.0914)0.8524 (0.1056)0.2493 (0.0765)0.5741 (0.0969)0.4051 (0.0899)0.6218 (0.0964)(30,15,20)1MLE0.3482 (0.0877)0.6854 (0.0982)0.4979 (0.0923)0.8698 (0.1155)0.2567 (0.0759)0.5854 (0.0981)0.4209 (0.0877)0.6385 (0.0971)Bayes0.3341 (0.0775)0.6774 (0.0902)0.483 (0.0876)0.844 (0.0989)0.2457 (0.0664)0.5695 (0.0821)0.3962 (0.0812)0.6149 (0.0886)2MLE0.3476 (0.0873)0.6871 (0.984)0.4939 (0.0926)0.8679 (0.1157)0.2532 (0.0782)0.5845 (0.0981)0.4206 (0.0876)0.6395 (0.0978)Bayes0.3317 (0.0772)0.6715 (0.0903)0.4819 (0.0875)0.8475 (0.0985)0.2439 (0.0667)0.5642 (0.0822)0.3959 (0.0819)0.6139 (0.0883)3MLE0.3466 (0.0875)0.6853 (0.0989)0.4958 (0.0924)0.8688 (0.1163)0.2549 (0.0784)0.5835 (0.0987)0.4208 (0.0874)0.6393 (0.0973)Bayes0.3325 (0.0778)0.6703 (0.0901)0.4769 (0.0867)0.8395 (0.0987)0.2427 (0.0667)0.5624 (0.0821)0.3955 (0.0818)0.6129 (0.0886)(40,15,20)1MLE0.3355 (0.0812)0.679 (0.0924)0.4864 (0.0856)0.8526 (0.1021)0.2508 (0.0719)0.5712 (0.0887)0.4083 (0.0815)0.6296 (0.0876)Bayes0.3255 (0.0702)0.6671 (0.0833)0.4681 (0.0756)0.8358 (0.0914)0.2417 (0.0603)0.5546 (0.0767)0.386 (0.0744)0.6025 (0.0807)2MLE0.3358 (0.0814)0.6755 (0.0914)0.4822 (0.0851)0.8588 (0.1026)0.2445 (0.0718)0.573 (0.0878)0.4077 (0.0814)0.6305 (0.0881)Bayes0.3254 (0.0706)0.6644 (0.0839)0.4669 (0.0756)0.8354 (0.0914)0.2312 (0.0605)0.5539 (0.0766)0.3856 (0.0748)0.6022 (0.0809)3MLE0.3368 (0.0818)0.6768 (0.0915)0.4865 (0.0855)0.8577 (0.1027)0.2448 (0.0713)0.5721 (0.0873)0.4091 (0.0812)0.6317 (0.0891)Bayes0.3256 (0.0708)0.668 (0.0837)0.4686 (0.0758)0.833 (0.0911)0.2348 (0.0607)0.5537 (0.0763)0.3879 (0.0748)0.6027 (0.0811)(40,20,25)1MLE0.3229 (0.0748)0.6655 (0.0836)0.4721 (0.0768)0.6633 (0.0941)0.246 (0.0661)0.4029 (0.0803)0.2338 (0.0777)0.471 (0.0802)Bayes0.3123 (0.0644)0.6523 (0.0765)0.4579 (0.0675)0.8257 (0.0885)0.2335 (0.0579)0.5483 (0.0696)0.3727 (0.0705)0.6018 (0.0731)2MLE0.3213 (0.0746)0.6656 (0.0833)0.473 (0.0766)0.8479 (0.0949)0.2365 (0.0669)0.5621 (0.0801)0.3973 (0.0772)0.6262 (0.0808)Bayes0.3127 (0.0642)0.6502 (0.0762)0.456 (0.0674)0.8259 (0.0888)0.2247 (0.0577)0.5489 (0.0697)0.3737 (0.0701)0.6019 (0.0729)3MLE0.3215 (0.0746)0.6625 (0.0839)0.4767 (0.0767)0.6692 (0.0952)0.2361 (0.0662)0.5628 (0.0809)0.2324 (0.0774)0.6212 (0.0801)Bayes0.3141 (0.0641)0.651 (0.0761)0.4563 (0.0679)0.8232 (0.0887)0.2241 (0.0573)0.5479 (0.0699)0.3727 (0.0703)0.5931 (0.0731)(50,20,25)1MLE0.3016 (0.0621)0.6596 (0.0751)0.4641 (0.0656)0.8265 (0.0852)0.2343 (0.0576)0.5522 (0.0709)0.3852 (0.0675)0.6114 (0.0707)Bayes0.2924 (0.0549)0.6395 (0.0674)0.4429 (0.0567)0.8187 (0.0779)0.2271 (0.0509)0.5319 (0.0623)0.364 (0.0605)0.5807 (0.0629)2MLE0.3017 (0.0627)0.6587 (0.0748)0.3418 (0.0657)0.8281 (0.0851)0.2248 (0.0582)0.5515 (0.0706)0.3879 (0.0674)0.6114 (0.0711)Bayes0.2916 (0.0549)0.6387 (0.0679)0.4457 (0.0562)0.8152 (0.0778)0.2174 (0.0505)0.5329 (0.0619)0.3644 (0.0607)0.5832 (0.0621)3MLE0.3093 (0.0626)0.6537 (0.0749)0.4651 (0.0655)0.8274 (0.0859)0.2269 (0.0577)0.5518 (0.0701)0.3852 (0.0679)0.6103 (0.0712)Bayes0.297 (0.0544)0.6359 (0.0677)0.4455 (0.0564)0.8176 (0.0781)0.2169 (0.0507)0.5325 (0.0621)0.3628 (0.0603)0.5846 (0.0624)(50,25,30)1MLE0.2828 (0.0508)0.6414 (0.0639)0.4527 (0.0511)0.8164 (0.0761)0.2251 (0.0471)0.5362 (0.0607)0.3724 (0.0511)0.6059 (0.0681)Bayes0.2752 (0.0407)0.6345 (0.0563)0.4318 (0.0507)0.8033 (0.0688)0.2169 (0.0404)0.4379 (0.0508)0.3552 (0.0479)0.5742 (0.0524)2MLE0.2892 (0.0509)0.6423 (0.0637)0.4517 (0.0507)0.8151 (0.0768)0.2163 (0.0478)0.5391 (0.0611)0.3747 (0.0518)0.4871 (0.0678)Bayes0.273 (0.0403)0.6233 (0.0561)0.4312 (0.0509)0.8016 (0.0685)0.2046 (0.0405)0.5216 (0.0509)0.3585 (0.0482)0.5762 (0.0531)3MLE0.2851 (0.0509)0.6443 (0.0635)0.4508 (0.0505)0.815 (0.0769)0.2152 (0.0477)0.5377 (0.0609)0.3755 (0.0517)0.6072 (0.0682)Bayes0.2716 (0.0406)0.6276 (0.0558)0.4352 (0.0506)0.803 (0.0687)0.2062 (0.0407)0.5224 (0.0507)0.3595 (0.0481)0.5766 (0.0527)Table 4
The CPs and AILs of the confidence intervals for unknown parameters (τ=0.5).
n,m1,m2SchemeCP (AIL)λ1α10α11α12λ2α20α21α22(30,10,15)1MLE84.7 (0.2892)84.7 (0.3312)84.4 (0.2763)85.1 (0.3762)85.2 (0.2195)84.6 (0.2953)84.7 (0.2682)85.8 (0.3152)Bayes88.1 (0.2768)87.1 (0.3198)88.1 (0.2684)88.7 (0.3638)89.1 (0.2079)90.2 (0.2861)88.4 (0.2586)89.3 (0.3055)2MLE84.3 (0.2868)84.5 (0.3237)85.9 (0.2748)85.2 (0.3724)85.1 (0.2174)84.2 (0.2972)84.5 (0.2684)85.3 (0.3154)Bayes87.9 (0.2732)88.2 (0.3158)89.2 (0.2681)88.7 (0.3628)89.5 (0.2738)90.1 (0.2822)88.2 (0.2558)89.1 (0.3037)3MLE84.2 (0.2872)84.1 (0.3254)85.1 (0.2775)85 (0.3713)85.2 (0.2174)84.9 (0.2961)84.4 (0.2653)85.3 (0.3171)Bayes87.2 (0.2752)88.1 (0.3149)89.1 (0.2669)88.1 (0.3652)89.4 (0.2044)89.9 (0.2846)88.2 (0.2393)89.2 (0.3024)(30,15,20)1MLE86 (0.2675)86.3 (0.3084)85.2 (0.2569)86.5 (0.3546)87 (0.1989)85.2 (0.2773)86.8 (0.2489)87 (0.2965)Bayes90.3 (0.2546)89.2 (0.2969)89.6 (0.2478)89.2 (0.3412)90.3 (0.1891)91.1 (0.2684)89.9 (0.2372)91.1 (0.2858)2MLE85.6 (0.2672)85.7 (0.3058)86.3 (0.2545)87.2 (0.3553)86.7 (0.1987)86.2 (0.2785)85.8 (0.2487)87.6 (0.2966)Bayes89.6 (0.2549)90.3 (0.2964)89.4 (0.2469)89.3 (0.3454)90.2 (0.1871)91.8 (0.2684)89.2 (0.2379)90.2 (0.2845)3MLE85.4 (0.2689)85.6 (0.3074)86.4 (0.2582)86 (0.3591)87 (0.2025)86.2 (0.2778)85.6 (0.2483)86.8 (0.2954)Bayes89.1 (0.2571)89.7 (0.2956)90.1 (0.2448)89.3 (0.3447)90.4 (0.1887)90.1 (0.2676)89.3 (0.2368)90.2 (0.2822)(40,15,20)1MLE87.4 (0.2527)87.7 (0.2873)86.1 (0.2366)86.8 (0.3362)88 (0.1831)87.3 (0.2569)88 (0.2287)89 (0.2778)Bayes91.4 (0.2376)92.1 (0.2765)92.9 (0.2264)91.1 (0.3254)91.6 (0.1683)92.6 (0.2488)92 (0.2185)92.3 (0.2687)2MLE87 (0.2545)86.6 (0.2853)88.4 (0.2347)88.5 (0.3376)88.3 (0.1826)88.8 (0.2595)87.4 (0.2273)89.4 (0.2795)Bayes91.5 (0.2423)92.3 (0.2733)91.6 (0.2276)91.6 (0.3262)91.6 (0.1667)92.2 (0.2457)91.5 (0.2177)92.8 (0.2693)3MLE87.1 (0.2579)86.9 (0.2893)87.9 (0.2376)88.1 (0.3368)88 (0.1852)88.1 (0.2548)87.2 (0.2297)88.7 (0.2762)Bayes91.1 (0.2425)92 (0.2758)91.2 (0.2269)91.1 (0.3245)91.5 (0.1685)92.2 (0.2481)91.3 (0.2183)92.3 (0.2679)(40,20,25)1MLE89 (0.2333)89 (0.2736)88 (0.2149)88.3 (0.3258)90 (0.1684)89.6 (0.2335)90 (0.2142)90.2 (0.2611)Bayes94.2 (0.2273)94.1 (0.2543)94.5 (0.2076)93.5 (0.3062)94.1 (0.1482)94.2 (0.2246)93.3 (0.1964)93.2 (0.2563)2MLE89.5 (0.2362)89 (0.2698)89.5 (0.2182)89.5 (0.3221)89.2 (0.1651)89.8 (0.2361)89 (0.2138)91 (0.2578)Bayes94.4 (0.2256)94.9 (0.2577)94.8 (0.2086)94.3 (0.3076)93.9 (0.1463)94.8 (0.2265)93.5 (0.1967)94.4 (0.2467)3MLE89.3 (0.2344)88.2 (0.2756)89.2 (0.2226)89.3 (0.3221)89.7 (0.1692)89.7 (0.2368)88.7 (0.2139)89.9 (0.2572)Bayes94.3 (0.2272)94.8 (0.2566)94 (0.2077)94.3 (0.3089)93.3 (0.1469)94.1 (0.2344)93.3 (0.1987)94.1 (0.2452)(50,20,25)1MLE90.9 (0.2166)89.9 (0.2547)89.9 (0.2051)89.5 (0.3089)90.8 (0.1474)90.8 (0.2148)91.8 (0.1989)91.3 (0.248)Bayes95.7 (0.2066)96.1 (0.2342)95.6 (0.1878)95 (0.2854)96.8 (0.1324)95.9 (0.2078)95.2 (0.1732)95.1 (0.2283)2MLE90.8 (0.2185)89.2 (0.2532)90.7 (0.2025)90.2 (0.3097)90.9 (0.1442)90.1 (0.2176)90.8 (0.1942)91.7 (0.2423)Bayes95.2 (0.2068)96.2 (0.2338)95.8 (0.1853)97.8 (0.2887)96.2 (0.1292)96.7 (0.2083)97.8 (0.1784)96.4 (0.2256)3MLE90.5 (0.2169)89.1 (0.2541)90.2 (0.2015)90.1 (0.3076)90.3 (0.1458)90.1 (0.2234)90.1 (0.1935)91.1 (0.2373)Bayes95.1 (0.2089)96.1 (0.2429)95.2 (0.1871)97 (0.2895)96.4 (0.1282)96.3 (0.2143)96.8 (0.1763)96.3 (0.2245)(50,25,30)1MLE92.7 (0.1967)91.7 (0.2357)91.6 (0.1879)92.1 (0.2865)91.5 (0.1263)91.6 (0.1971)92.4 (0.1784)93 (0.2266)Bayes97.1 (0.1866)97.8 (0.2151)97.7 (0.1657)97.9 (0.2648)98.3 (0.1132)97.9 (0.1843)98.4 (0.1579)97.8 (0.2058)2MLE91.6 (0.1971)91.4 (0.2356)92.6 (0.1857)92.2 (0.2859)92.5 (0.1287)92.4 (0.1954)92.8 (0.1784)93.5 (0.2283)Bayes97.6 (0.1832)97.3 (0.2152)97.5 (0.1669)98 (0.2666)98.6 (0.1187)97.5 (0.1824)98.7 (0.1576)97.8 (0.2073)3MLE91.4 (0.1978)91.2 (0.2352)92.1 (0.1834)92.2 (0.2858)92.1 (0.1266)92.2 (0.2026)92.1 (0.1728)93.2 (0.2134)Bayes97.5 (0.1852)97.1 (0.2276)97.9 (0.1653)98.1 (0.2673)97.5 (0.1142)97.1 (0.1945)98 (0.1575)97.8 (0.2065)Table 5
The CPs and AILs of the confidence intervals for unknown parameters (τ=1).
n,m1,m2SchemeCP (AIL)λ1α10α11α12λ2α20α21α22(30,10,15)1MLE84.7 (0.2886)84.3 (0.3294)84.3 (0.2739)84.9 (0.3737)85.2 (0.2186)84.5 (0.2922)84.3 (0.2662)85.1 (0.3135)Bayes88.1 (0.2737)87.3 (0.3177)87.9 (0.2675)88.4 (0.3619)88.9 (0.2056)90.2 (0.2832)88.4 (0.2563)89.1 (0.3027)2MLE84.5 (0.2875)84.2 (0.3246)84.3 (0.2723)84.6 (0.3732)85.1 (0.2168)84.2 (0.2952)84.1 (0.269)84.8 (0.3172)Bayes88.1 (0.2755)87.1 (0.3168)87.6 (0.2659)88.2 (0.3625)88.2 (0.2027)89.7 (0.2821)88.2 (0.2528)89.2 (0.3018)3MLE84.3 (0.2881)84.2 (0.3239)84.2 (0.2751)84.4 (0.3702)85 (0.2189)84.1 (0.297)84.2 (0.2611)84.6 (0.3138)Bayes88.1 (0.2748)87.3 (0.3127)87.4 (0.2666)88.1 (0.3622)88.1 (0.1989)89.5 (0.2812)88.3 (0.2532)89.1 (0.3015)(30,15,20)1MLE85.7 (0.2694)86.2 (0.3076)85.8 (0.2577)86.2 (0.3544)86.5 (0.1999)86.2 (0.2785)85.3 (0.2478)86.9 (0.2948)Bayes90.5 (0.2569)89.1 (0.2965)89.6 (0.2485)89.1 (0.3426)90.1 (0.1885)90.1 (0.2676)89.4 (0.2388)90.1 (0.2869)2MLE85.5 (0.2684)86.1 (0.3065)85.7 (0.2588)86.4 (0.3589)86.4 (0.1997)86.1 (0.2797)86 (0.2481)86.2 (0.2987)Bayes90.2 (0.2596)89.3 (0.2974)89.6 (0.2486)89 (0.3456)90 (0.1864)90.1 (0.2679)89.3 (0.2356)90.1 (0.2852)3MLE85.4 (0.2702)86.1 (0.3096)85.7 (0.2571)86.2 (0.3605)89.3 (0.2041)86.2 (0.2798)85.8 (0.2495)86.2 (0.2962)Bayes90 (0.2583)89.2 (0.2977)89.3 (0.2463)88.8 (0.3414)89.9 (0.1887)90.1 (0.2675)89.2 (0.2369)90.2 (0.283)(40,15,20)1MLE87.2 (0.2513)87.3 (0.2888)86.7 (0.2369)86.7 (0.3377)87.9 (0.1812)87.3 (0.2589)87.6 (0.2277)88.5 (0.2765)Bayes91.2 (0.2426)91.6 (0.2713)92.6 (0.2296)91 (0.3249)91.4 (0.1697)92.4 (0.2484)91.9 (0.2197)92.2 (0.2696)2MLE87.2 (0.2537)87.2 (0.2876)86.3 (0.2318)87.8 (0.3399)87.8 (0.1807)87.2 (0.2596)87.5 (0.2298)88.2 (0.2787)Bayes91.1 (0.2419)91.2 (0.2737)92.7 (0.2278)91 (0.3232)91.2 (0.1687)92.4 (0.2437)91.7 (0.2189)92 (0.2683)3MLE87 (0.2562)87.1 (0.2908)86.1 (0.2396)87.7 (0.3356)87.8 (0.1874)87.2 (0.2585)87.5 (0.2303)88.2 (0.2785)Bayes90.9 (0.2412)91.2 (0.2765)92.1 (0.2257)90.8 (0.3275)91.2 (0.1667)92.2 (0.2493)91.7 (0.2141)92 (0.2625)(40,20,25)1MLE88.6 (0.2346)88.9 (0.2724)87.9 (0.2184)88.2 (0.3212)89.5 (0.1686)89.4 (0.2342)89.1 (0.2139)90.1 (0.2601)Bayes93.8 (0.2265)94 (0.2584)94.3 (0.2093)93.2 (0.3086)94.1 (0.1499)94.1 (0.2253)93.7 (0.1997)94 (0.2579)2MLE88.4 (0.2369)88.7 (0.2692)87.9 (0.2193)88.3 (0.3215)89.2 (0.1677)89.3 (0.2372)89.1 (0.2104)90 (0.2597)Bayes93.8 (0.2273)93.9 (0.2595)94.4 (0.2091)93.1 (0.3093)93.9 (0.1476)94 (0.2295)93.4 (0.1999)93.8 (0.2448)3MLE88.4 (0.235)88.7 (0.2719)87.8 (0.2219)88 (0.3241)89.2 (0.1685)89.3 (0.2393)89 (0.2123)89.6 (0.2587)Bayes93.4 (0.2286)93.9 (0.2598)94.2 (0.2085)93.3 (0.3108)93.8 (0.1439)93.8 (0.2314)93.3 (0.1979)93.6 (0.2436)(50,20,25)1MLE90.5 (0.2164)89.6 (0.2561)89.6 (0.2048)90.1 (0.3099)90.3 (0.1474)90.5 (0.2186)91.4 (0.1964)91.3 (0.2486)Bayes95.4 (0.2072)96.1 (0.2337)95.6 (0.1853)95.1 (0.2876)96.3 (0.1314)95.4 (0.2089)95.1 (0.1776)95 (0.2243)2MLE90.3 (0.2195)89.5 (0.2517)89.5 (0.2015)90 (0.3093)90.1 (0.1481)90.1 (0.2192)91.4 (0.1947)91.2 (0.2401)Bayes95.4 (0.2095)95.8 (0.2359)95.3 (0.1877)95 (0.2892)96.2 (0.1287)95.3 (0.2107)95.1 (0.1795)95 (0.2261)3MLE90.2 (0.2167)89.5 (0.2538)89.4 (0.2029)89.5 (0.3085)89.7 (0.1497)90.1 (0.2213)91.3 (0.1942)91.2 (0.2389)Bayes94.3 (0.2104)95.3 (0.2413)95.2 (0.1895)95 (0.2886)96.1 (0.1274)95.2 (0.2171)95.1 (0.1778)94.9 (0.2216)(50,25,30)1MLE92.7 (0.1987)91.4 (0.2356)91.3 (0.1864)92.6 (0.2885)91.4 (0.1237)92.3 (0.1993)92.3 (0.1763)93 (0.224)Bayes97.1 (0.1857)97.3 (0.2134)97.2 (0.1646)97.2 (0.2661)98.2 (0.1114)97.5 (0.1856)98.4 (0.1575)97.8 (0.2037)2MLE92.6 (0.1972)91.4 (0.2364)91.1 (0.1876)92.4 (0.2869)91.4 (0.1261)92.2 (0.1927)92.2 (0.1769)92.8 (0.2222)Bayes97.1 (0.1829)97.3 (0.2155)97.2 (0.1655)97.1 (0.2658)98.2 (0.1142)97.4 (0.1887)98.3 (0.1586)97.8 (0.2062)3MLE92.6 (0.1957)91.4 (0.2357)91.1 (0.1835)92.4 (0.2886)91.4 (0.1274)92.2 (0.2013)92.2 (0.1733)92.8 (0.2162)Bayes97.8 (0.1848)97.3 (0.2258)97.2 (0.1642)97.1 (0.2679)98.2 (0.1145)97.4 (0.1923)98.3 (0.1566)97.8 (0.2045)From Tables2 and 3, it can be observed that the MSEs of the MLEs and BEs for the unknown parameters are decreased with the sample size n increase or the effective sample size m increases, and the MLEs and BEs of the unknown parameters are closer to the true values. For the bigger τ, the MSEs for the MLEs and Bayes of unknown parameters both are the smaller. In addition, the difference under three difference progressive censored schemes is not sensible. Furthermore, the BEs of the unknown parameters are better than MLEs, and its MSEs are always less than the MLEs.From Tables4 and 5, the CPs of ACIs and HPD CIs for the unknown parameters are increased, and meanwhile, the AILs of the ACIs and the HPD CIs are decreased with the sample size n or the effective sample size m increase. For the fixed n, m, and τ, the CPs of HPD CIs are more stable than the ACIs for the unknown parameters.


## 5.2. Real Data Analysis

To illustrate the proposed methods, a real data is selected from Ebrahimi [25], of which the accelerated stress is voltage, the stress levels are S1=10V,S2=15V,andS3=20V, and the normal stress level is S0=5V. Furthermore, for each stress level, the life tests are allowed to run over the time τ1,τ2,τ3=2,1.5,1, the numbers under the test were n1,n2,n3=15,15,15, the expected failure number are m1,m2,m3=11,12,13, and the progressive censoring scheme is R1=R2=⋯=Rmi−1=0andRmi=ni−mi,i=1,2,and3.Assume that the unknown failure cause of the failure data proportion is 20%, and then, the failure data and the corresponding failure causes under each stress level are presented in Table6.Table 6
The failure data with the dependent competing risks’ model.
S1(0.04, 0)(0.10, 1)(0.16, 0)(0.17,∗)(0.42, 2)(0.46, 0)(0.47,∗)(0.47, 1)(1.05, 1)(1.50, 2)(1.80, 0)S2(0.02, 1)(0.06, 2)(0.27,∗)(0.30, 2)(0.30, 2)(0.34, 1)(0.36,∗)(0.70, 2)(0.80, 2)(0.80, 0)(0.92,∗)(1.06, 2)S3(0.07, 2)(0.13, 1)(0.20,∗)(0.29, 1)(0.30, 2)(0.30,∗)(0.30, 1)(0.33, 2)(0.33,∗)(0.40, 0)(0.40, 1)(0.63,∗)(0.82, 0)For the Bayes analysis, we select small values of the hyperparametersci0=ci1=ci2=di0=di1=di2=0.0001, which gives noninformation priors [26]. We make the data in Table 6 be brought to the above mode and obtain the point estimation and interval estimation of the unknown parameters. The results are shown in Table 7.Table 7
Estimations and 95% confidence and incredible intervals.
EstimatorMLEACIBayesHPDλ10.3133(0.0911, 0.5982)0.3027(0.0953, 0.5976)α101.7241(0.9851, 2.4724)1.7163(0.9735, 2.3853)α110.8616(0.3953, 1.1285)0.8473(0.4075, 1.1153)α120.8247(0.4261, 1.1176)0.8193(0.4378, 1.1104)λ20.8359(0.3956, 1.1432)0.8287(0.3918, 1.1427)α201.4937(0.6853, 2.2589)1.4517(0.6978, 2.2667)α210.7745(0.4137, 1.0134)0.7689(0.4166, 1.0089)α221.1201(0.6176, 1.6792)1.0816(0.6186, 1.6811)λ31.1024(0.5983, 1.5925)1.0673(0.6017, 1.5937)α302.3691(1.4327, 3.2896)2.2164(1.4673, 3.2576)α311.4151(0.8221, 2.1074)1.4091(0.8226, 2.0974)α321.5121(0.8831, 2.2146)1. 4973(0.8875, 2.2121)From Table7, we can obtain that there is little difference between the MLEs and BEs of the unknown parameters. The lengths for the HPD CIs are shorter than the ACIs in most cases. For the given significance value, the accuracy of the HPD CIs is better than the ACIs.Here, aiming at the above data analysis, there exists a question on how good the proposed MOBG model fits the data. That is, there have been multiple goodness-of-fit testing methods for univariate distributions. However, for bivariate distribution, the goodness-of-fit testing methods are less. Thus, to verify the above fit results of data and distribution, we mainly used the method which is from the Kundu and Gupta [27], and we verified two margins and the minimum lifetime by using the Kolmogorov–Smirnov test (K–S test).Give the significance valueγ=5%, the K–S statistic and the corresponding p values under different failure numbers are obtained, and the result is presented in Table 8. The comparison charts for empirical distribution and its fitted distribution of the sample under each stress are shown in Figures 1–3.Table 8
The K–S statistics and the correspondingp value.
DistributionFt;λ1,α101Ft;λ1,α102Ft;λ1,α1Ft;λ2,α201Ft;λ2,α202Ft;λ2,α2Ft;λ3,α301Ft;λ3,α302Ft;λ3,α3K–S0.24430.14350.16890.15150.17150.20350.17370.20890.2388p0.55620.95360.86200.90760.81570.63250.76760.63240.5873Figure 1
Comparison chart for empirical distribution and its fitted distribution underS1 level. (a) The comparison chart when α=α101. (b) The comparison chart when α=α102. (c) The comparison chart when α=α1.
(a)(b)(c)Figure 2
Comparison chart for empirical distribution and its fitted distribution underS2 level. (a) The comparison chart when α=α201. (b) The comparison chart when α=α202. (c) The comparison chart when α=α2.
(a)(b)(c)Figure 3
Comparison chart for empirical distribution and its fitted distribution underS3 level. (a) The comparison chart when α=α301. (b) The comparison chart when α=α302. (c) The comparison chart when α=α3.
(a)(b)(c)From Table8, we can obtain that all the corresponding p values are bigger than the given significance value. It indicates that the failure data comes from the MOBG distribution.


## 6. Conclusion

Under the AT-II PHCS, we consider statistical analysis of the dependent competing risk model by using the MOBG distribution under CSALT in which the shape parameters of life distribution are changed with the stress levels. Moreover, we also consider the failure causes of some units are unknown in the dependent competing risks’ model, and then, both the ML method and Bayes approach are used to obtain the estimations of the unknown parameters. Based on these works, the expected values and MSEs of MLEs and BEs are derived, and the AILs and CPs of the unknown parameters based on ACIs and HPD CIs are also obtained. Finally, the performances of all procedures are evaluated. The simulation results show that the MSEs, AIL, and CPs of the unknown parameters perform well, which are in the bigger sample sizen or effective sample size m. Furthermore, compared with the ML method, the Bayes approach is superior according to the MSEs, AIL, and CPs of the unknown parameters.



---

*Source: 6641859-2021-04-16.xml*