# Bayesian Inference for Step-Stress Partially Accelerated Competing Failure Model under Type II Progressive Censoring

Authors: Xiaolin Shi; Fen Liu; Yimin Shi
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper deals with the Bayesian inference on step-stress partially accelerated life tests using Type II progressive censored data in the presence of competing failure causes. Suppose that the occurrence time of the failure cause follows Pareto distribution under use stress levels. Based on the tampered failure rate model, the objective Bayesian estimates, Bayesian estimates, and E-Bayesian estimates of the unknown parameters and acceleration factor are obtained under the squared loss function. To evaluate the performance of the obtained estimates, the average relative errors (AREs) and mean squared errors (MSEs) are calculated. In addition, the comparisons of the three estimates of unknown parameters and acceleration factor for different sample sizes and different progressive censoring schemes are conducted through Monte Carlo simulations.


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## Body

## 1. Introduction

The competing failure model is commonly used in medical studies or reliability analysis. Due to the complexity in the internal structure and the external working environment, there are often many causes of the product failure. Each of these causes is possible to result in final failure of the product. This is called competing failure. In recent years, some of authors have investigated the competing failure models. Han and Kundu [1] studied the statistical analysis of the step-stress accelerated life test (SSALT) with competing risks for failure from the generalized exponential distribution under Type I censoring, and the point estimation and confidence interval of the distribution parameters are obtained. In the case that failure follows the lognormal distribution, Roy and Mukhopadhyay [2] investigated the maximum likelihood analysis of the accelerated life test of competing failure products using the EM algorithm. Zhang et al. [3] applied the copula model to combine the marginal distribution of the competing failure models with the time to failure distribution of the products and derived a general statistical model for accelerated life test (ALT) with s-dependent competing failures. Shi et al. [4] considered a constant-stress accelerated life test (CSALT) with competing risks for failure from exponential distribution under progressive Type II hybrid censoring. The maximum likelihood estimator and Bayes estimator of the parameter were derived. Xu and Tang [5] and Wu et al. [6] considered different inferential issues regarding the constant-stress accelerated competing failure models when the lifetime of different risk factors follows Weibull distribution. Other research results can be found in [7, 8].The above references considered competing failure models in the case of CSALT or SSALT. Based on the CSALT or SSALT to analyze the life characteristics of product, we need to use the relationship between product life and stress levels, that is, acceleration model. But, in engineering practice, the accelerated model is not always known, such as the newly developed products. In order to solve this problem, the partially accelerated life test (PALT) is more suitable to be performed. There are two types of PALT, namely, constant-stress PALT (CSPALT) and step-stress PALT (SSPALT). The latter allows for changing the stress from normal level to accelerated level at a specific time point, which contributes to getting the information of the products’ lifetimes quickly. Under the PALT, Abd-Elfattah et al. [9] and Ismail [10, 11] derived the maximum likelihood estimates and Bayesian estimates of the acceleration factor and unknown parameters for different lifetime distributions. Abushal and Soliman [12] studied the MLE and Bayesian estimate of the parameters for the Pareto distribution of the second type under progressive censoring data for constant PALT. Ismail and Sarhan [13] discussed optimal design of step-stress life test with progressively Type II censored exponential data. But few authors have studied competing failure models from this perspective and especially from the Bayesian viewpoint.The Bayesian method has certain accurate advantages when sample size is small. When little knowledge about the prior information is obtained, we can derive the objective Bayesian estimates based on noninformative priors, such as the Jeffreys prior and reference prior. For more details, see Jeffreys [14]. Guan et al. [15] derived the objective Bayesian estimates of unknown parameters for bivariate Marshall-Olkin exponential distribution, respectively. Under Type I censoring, Xu and Tang [5] considered the objective Bayesian inference for the accelerated competing failure models. Instead, with historical data or experience of some experts available, the subjective Bayesian approach on the basis of informative prior distributions can be implemented, while if the hyperparameters of the informative prior distributions are unknown, we are able to use E-Bayesian method proposed by Han [16, 17]. By assuming the hyperparameters follow certain prior distributions, the E-Bayesian estimate can be obtained as the expectation of the Bayesian estimate of the unknown parameter.In life testing and reliability studies, the experimenter may not always obtain complete information on failure times for all experimental units. Thus, censoring is common to be performed. A censoring scheme (CS), which can balance between total time spent for the experiment, number of units used in the experiment, and the efficiency of statistical inference based on the results of the experiment, is desirable [12]. Compared to the conventional Type I and Type II censoring scheme, Type II progressive censoring scheme provides higher flexibility to the experimenter by allowing for product to be removed at nonterminal time points. This allowance may be desirable when some degradation-related information from live test specimens needs to be collected and analyzed or when scarce testing facilities need to be released for other experimentation. In recent years, some of researchers engaged in reliability research using progressively censored data, such as Abushal and Soliman [12] and Ismail and Sarhan [13].The Pareto distribution has found widespread use as a model for various socioeconomic phenomena; see [18, 19]. The Pareto distribution has also been used in reliability and lifetime modelling (see [20–22]). In this paper, we will consider Bayesian analysis for the step-stress partially accelerated competing failure model from Pareto distribution under Type II progressive censoring. The rest of this paper is organized as follows. In Section 2, under Type II progressive censoring scheme, a step-stress partially accelerated competing failure model from Pareto distribution is described and some basic assumptions are given. In Section 3, we obtain the objective Bayesian estimates of the acceleration factor and unknown parameters based on the noninformative prior distribution. Furthermore, the Bayesian and corresponding E-Bayesian estimates are provided. In Section 4, we provide two real data sets to justify the use of the Pareto distribution in engineering practice. The simulation results of all proposed methods for different sample sizes and for different progressive censoring schemes are presented in Section 5, and some conclusions are presented in Section 6.


## 2. Assumptions and Model Description

Throughout the paper, we make the following basic assumptions:(1)
The failure of a product occurs only due to one of the 2 independent competing failure causes with lifetimesT1 and T1. Then the lifetime of the product is T=min⁡(T1,T2).(2)
Two stress levelsS0 and S1 are used in SSPALT (S0<S1), where S0 is use stress level and S1 is accelerated stress level.(3)
LetT1j be the occurrence time of the jth failure cause under stress levels S0 and T2j be the occurrence time of the jth failure cause under stress levels S1, j=1,2. The times T1j and T2j are independent. Also, T1j follows a Pareto distribution with the following hazard rate function (HRF) and reliability function: (1)r1jt=αjt,t≥θj,S1jt=θjαjt-αj,t≥θj,j=1,2,whereθj>0 is the scale parameter and αj>0 is the shape parameter, j=1,2.(4)
Tampered failure rate model holds: that is, the HRFr1j is proportional to r2j. Thus, the HRF of thejth failure cause in SSPALT can be expressed as(2)rj∗t=r1jt=αjt,θj≤t≤τ,r2jt=αjλt,t>τ,λ>1,whereτ is prefixed number, τ∈(0,∞), and λ is acceleration factor. The corresponding reliability function is(3)Sj∗t=S1jt=θjαjt-αj,θj≤t≤τ,S2jt=θjαjταjλ-1t-αjλ,t>τ,λ>1.Under Type II progressive censored scheme, SSPALT is described as follows: n identical products are placed on the test under the stress level S0. At the time of the first failure, R1 products are randomly removed from the remaining n-1 products, and the failure cause index c1 is recorded. Similarly, at the time of the second failure, R2 products are randomly removed from the remaining n-2-R1 products, and the failure cause index c2 is recorded. The test continues until time point τ, the stress levels are changed from S0 to S1. Under the stress level S1, the remaining n-N1-R1-R2-⋯-RN1 products continue to be tested. Suppose that the total number of failures under the stress level S0 is N1. At the time of the (N1+1)th failure, RN1+1 products are randomly removed from the remaining n-(N1+1)-R1-R2-⋯-RN1 products, and the failure cause index cN1+1 is recorded. The test continues until themth failure is observed and the failure cause index cm is recorded. At this time, all the remaining Rm=n-m-R1-R2-⋯-Rm-1 products are removed and the test terminates, where m, Ri (i=1,2,…,m-1) are prefixed number (N1<m<n) and ci∈{1,2}, i=1,2,…,m. We denote the time of theith failure by ti and assume that each failure time along with the failure cause index is observed, and there is at least one failure caused by thejth (j=1,2) failure cause under stress level Sk(k=0,1). Therefore, observed data can be denoted as t1,c1,t2,c2,…,(tm,cm). Then the likelihood function of the failure samples caused by the jth failure cause is(4)Lj∝∏i=1N1r1jδjcitiS1j1+Riti∏i=N1+1mr2jδjcitiS2j1+Riti=αjnjθjαjnλn2jU1jU2jW1-αjλW2αjλ-1,where(5)δjci=1,ci=j,0,ci≠j,U1j=∏i=1N1ti-δjci-αj1+Ri,U2j=∏i=N1+1mti-δjci,W1=∏i=N1+1mti1+Ri,W2=τAm,nj=n1j+n2j,n1j=∑i=1N1δjci,n2j=∑i=N1+1mδjci,Am=∑i=N1+1m1+Ri.Based on assumptions (1) and (4), we derive the full-likelihood function as(6)Lt∣λ,θ1,θ2=∏j=12Lj∝∏j=12αjnjU1jU2jθjαjnλm-N1W1-α1+α2λW2α1+α2λ-1.


## 3. Bayesian Analysis

In this section, the Bayesian method is employed to estimate the scale parameters and acceleration factor when the shape parameters are known. Depending on whether the prior information is available or not, we obtain the objective Bayesian estimates, Bayesian estimates, and E-Bayesian estimates of the unknown parameters and acceleration factor under the squared loss function, respectively.


### 3.1. Objective Bayesian Estimates

When little knowledge about the prior information is obtained, the objective Bayesian approach is an alternative way. We assume that the joint prior distribution of the parameters (λ,θ1,θ2) is(7)πoλ,θ1,θ2∝1λθ1θ2.From (6) and (7), the posterior density functions of the parameters λ, θ1, and θ2 can be given by πoλ,θ1,θ2∣t∝Lt∣λ,θ1,θ2πo(λ,θ1,θ2). Then the marginal posterior density functions of the parameters λ, θ1, and θ2 can be obtained, respectively:(8)πoλ∣t=D1λm-N1-1W1-α1+α2λW2α1+α2λ,λ>1,πoθ1∣t=D2θ1α1n-1,0<θ1<t11,πoθ2∣t=D3θ2α2n-1,0<θ2<t21,where tk1 is the first-failure time caused by failure cause k(k=1,2) and D1=[∫1∞λm-N1-1W1-(α1+α2)λW2(α1+α2)λdλ]-1, D2=[∫0t11θ1α1n-1dθ1]-1, and D3=[∫0t21θ2α1n-1dθ2]-1.Under the squared loss function, the objective Bayesian estimates of the parametersλ, θ1, and θ2 can be obtained as(9)λ^OBS=D1∫1∞λm-N1W1-α1+α2λW2α1+α2λdλ,θ^jOBS=αjntj1αjn+1-1,j=1,2.It is obvious to see that (9) does not have explicit form. So we obtain the Bayesian estimate of λ by numerical method. Let fλ=λm-N1-1W1-(α1+α2)λW2(α1+α2)λ, V=λα1+α2log⁡(W1/W2), and then(10)∫1∞λm-N1-1W1-α1+α2λW2α1+α2λdλ=∫0∞fλdλ-∫01fλdλ=α1+α2log⁡W1W2-m-N1∫0∞Vm-N1-1e-VdV-∫0V∗Vm-N1-1e-VdV=α1+α2log⁡W1W2-m-N1Γm-N1-∫0V∗Vm-N1-1e-VdV,whereV∗=α1+α2log⁡(W1/W2). Similarly we get(11)∫1∞λfλdλ=∫0∞λfλdλ-∫01λfλdλ=α1+α2log⁡W1W2-m-N1+1Γm-N1+1-∫0V∗Vm-N1e-VdV.Thus, under the squared loss function, the objective Bayesian estimates of the parameters λ can be obtained as(12)λ^OBS=D1∫1∞λm-N1W1-α1+α2λW2α1+α2λdλ=∫1∞λfλdλ∫1∞fλdλ=α1+α2log⁡W1/W2Γm-N1+1-∫0V∗Vm-N1e-VdVΓm-N1-∫0V∗Vm-N1-1e-VdV.About the integrals ∫0V∗Vm-N1e-VdV and ∫0V∗Vm-N1-1e-VdV, the compound trapezoidal integral formula can be used to obtain integral values. The specific steps are as follows: the interval [0,V∗] is divided into n equal subintervals and each length of subinterval is h=V∗/n. Then, from the compound trapezoidal integral formula, we get (13)∫0V∗gVdV≈h2g0+2∑k=1n-1gV∗kn+gV∗,∫0V∗fVdV≈h2f0+2∑k=1n-1fV∗kn+fV∗,where g(V)=Vm-N1e-V and f(V)=Vm-N1-1e-V. The above two integration results are substituted into (12), and the objective Bayesian estimates of the parameters λ can be obtained.


### 3.2. Bayesian Estimates

With historical data or experts’ experience available, the informative prior could be a better option. Following Doostparast et al. [23], a convenient prior family of distributions for θj(j=1,2) is provided by the power function distributions with probability density function(14)πsθj=βjθjβj-1dj-βj,0<θj<dj;dj,βj>0.We consider the prior density function ofλ as follows:(15)πsλ=Aλa-1exp⁡-bλ,a,b>0,λ>1,where A is the regularization factor.Then the marginal posterior density functions of the parametersλ, θ1, and θ2 are, respectively,(16)πsλ∣t∝λm-N1+a-1exp⁡-α1+α2ln⁡W1-ln⁡W2+bλ,a,b>0,λ>1,πsθj∣t=αjn+βjθjαjn+βj-1dj∗αjn+βj,0<θj<dj∗,dj∗=min⁡tj1,dj,where tj1 is the first-failure time caused by the jth failure cause (j=1,2).Under the squared loss function, the Bayesian estimate ofθj (j=1,2) can be easily obtained as(17)θ^jBS=∫0dj∗θjπSθj∣tdθj=αjn+βjdj∗αjn+βj+1.To obtain the Bayesian estimate ofλ, we expand H(λ)=λm-N1+a and h(λ)=λm-N1+a-1 in Taylor series around the maximum likelihood estimate λ^ to get the following approximation:(18)Hλ≈Hλ^+H′λ^λ-λ^,hλ≈hλ^+h′λ^λ-λ^,where H′(λ)=m-N1+aλm-N1+a-1 and h′λ=m-N1+a-1λm-N1+a-2.Therefore, under the squared loss function, the Bayesian estimate ofλ can be approximated as (19)λ^BS=Hλ^+H′λ^1+I1-1-λ^hλ^+h′λ^1+I1-1-λ^,where I1=(α1+α2)(ln⁡W1-ln⁡W2)+b.


### 3.3. E-Bayesian Estimates

By assuming that the hyperparameters follow certain distributions, the Bayesian estimates can be obtained when the hyperparameters in prior distributions are unknown. However, complicated integration is often a hard work. Han [16, 17] gave the definition of E-Bayesian estimation to overcome this problem. So we will derive the E-Bayesian estimates of the unknown parameters and acceleration factor in this subsection.Suppose that the hyperparametersa and b and βj and dj  (j=1,2) have the joint prior density functions πe(a,b) and πe(βj,dj), respectively. Then the E-Bayesian estimates of λ and θj  (j=1,2) are(20)λ^EB=∬Dλ^BSπea,bdadb,(21)θ^jEB=∬Dθ^jBSπeβj,djdβjddj,where λ^BS and θ^jBS are the Bayesian estimates of λ and θj  (j=1,2), respectively.According to Han [16], a and b and βj and dj  (j=1,2) should be selected to guarantee that πs(λ) and πs(θj) are decreasing functions of λ and θj, respectively. The derivatives of πs(λ) with respect to λ and πs(θj) with respect to θj are (22)∂πsλ∂λ∝λa-2exp⁡-bλa-1-bλ,∂πsθj∂θj=βjβj-1θjβj-2dj-βj,When 0<a<1, b>0, ∂πs(λ)/∂λ<0, and when 0<βj<1, dj>0, ∂πs(θj)/∂θj<0. Thus, πs(λ) and πs(θj) are decreasing functions of λ and θj, respectively.Given that0<a<1 and 0<βj<1, the larger b and dj are, the thinner the tail of πs(λ) and πs(θj) will be. Berger [24] showed that the thinner tailed prior distribution often reduces the robustness of the Bayesian estimate. Consequently, b and dj  (j=1,2) should be given upper bounds l and hj′, respectively. We consider the following two joint prior distributions of a and b and βj and dj  (j=1,2), respectively:(23)π1a,b=2bl2,0<a<1,0<b<l,π2a,b=1l,0<a<1,0<b<l,(24)π1jβj,dj=2djhj′2,0<βj<1,0<dj<hj′,π2jβj,dj=1hj′,0<βj<1,0<dj<hj′.From (19), (20), and (23), we derive the E-Bayesian estimates of λ as(25)λ^EB1=2l2∫0l∫01bHλ^+H′λ^1+I2-1-λ^hλ^+h′λ^1+I2-1-λ^dadb=2λ^l2∫0lb∫01I1+bλ^+m-N1+a1+I1+b1-λ^I1+bλ^+m-N1+a-11+I1+b1-λ^dadb=λ^+λ^l2Q1-Q2,λ^EB2=1l∫0l∫01Hλ^+H′λ^1+I2-1-λ^hλ^+h′λ^1+I2-1-λ^dadb=λ^l∫0l∫01I1+bλ^+m-N1+a1+I1+b1-λ^I1+bλ^+m-N1+a-11+I1+b1-λ^dadb=λ^+λ^lE1-E2,where(26)Q1=l2-I1+N2zλ^2ln⁡I1+lzλ^+N2+I1+N2zλ^2ln⁡I1zλ^+N2+N2lzλ^,Q2=l2-I1+N2-1pλ⌢2ln⁡I1+lpλ⌢+N2-1+I1+N2-1pλ⌢2ln⁡I1pλ⌢+N2-1+N2-1lpλ⌢,E1=I1+l+N2zλ^ln⁡I1+lzλ^+N2-I1+N2zλ^ln⁡I1zλ^+N2,E2=I1+l+N2-1pλ^ln⁡I1+lpλ^+N2-1-I1+N2-1pλ^ln⁡I1pλ^+N2-1,zλ^=λ^+N21-λ^,pλ^=λ^+N2-11-λ^,N2=m-N1.Similarly, from (17), (21), and (24), the E-Bayesian estimate of θj  (j=1,2) can be obtained as (27)θ^jEB1=2hj′31-ln⁡αjn+2αjn+1,dj∗=dj,dj∗1-ln⁡αjn+2αjn+1,dj∗≠dj,θ^jEB2=hj′21-ln⁡αjn+2αjn+1,dj∗=dj,dj∗1-ln⁡αjn+2αjn+1,dj∗≠dj.


## 3.1. Objective Bayesian Estimates

When little knowledge about the prior information is obtained, the objective Bayesian approach is an alternative way. We assume that the joint prior distribution of the parameters (λ,θ1,θ2) is(7)πoλ,θ1,θ2∝1λθ1θ2.From (6) and (7), the posterior density functions of the parameters λ, θ1, and θ2 can be given by πoλ,θ1,θ2∣t∝Lt∣λ,θ1,θ2πo(λ,θ1,θ2). Then the marginal posterior density functions of the parameters λ, θ1, and θ2 can be obtained, respectively:(8)πoλ∣t=D1λm-N1-1W1-α1+α2λW2α1+α2λ,λ>1,πoθ1∣t=D2θ1α1n-1,0<θ1<t11,πoθ2∣t=D3θ2α2n-1,0<θ2<t21,where tk1 is the first-failure time caused by failure cause k(k=1,2) and D1=[∫1∞λm-N1-1W1-(α1+α2)λW2(α1+α2)λdλ]-1, D2=[∫0t11θ1α1n-1dθ1]-1, and D3=[∫0t21θ2α1n-1dθ2]-1.Under the squared loss function, the objective Bayesian estimates of the parametersλ, θ1, and θ2 can be obtained as(9)λ^OBS=D1∫1∞λm-N1W1-α1+α2λW2α1+α2λdλ,θ^jOBS=αjntj1αjn+1-1,j=1,2.It is obvious to see that (9) does not have explicit form. So we obtain the Bayesian estimate of λ by numerical method. Let fλ=λm-N1-1W1-(α1+α2)λW2(α1+α2)λ, V=λα1+α2log⁡(W1/W2), and then(10)∫1∞λm-N1-1W1-α1+α2λW2α1+α2λdλ=∫0∞fλdλ-∫01fλdλ=α1+α2log⁡W1W2-m-N1∫0∞Vm-N1-1e-VdV-∫0V∗Vm-N1-1e-VdV=α1+α2log⁡W1W2-m-N1Γm-N1-∫0V∗Vm-N1-1e-VdV,whereV∗=α1+α2log⁡(W1/W2). Similarly we get(11)∫1∞λfλdλ=∫0∞λfλdλ-∫01λfλdλ=α1+α2log⁡W1W2-m-N1+1Γm-N1+1-∫0V∗Vm-N1e-VdV.Thus, under the squared loss function, the objective Bayesian estimates of the parameters λ can be obtained as(12)λ^OBS=D1∫1∞λm-N1W1-α1+α2λW2α1+α2λdλ=∫1∞λfλdλ∫1∞fλdλ=α1+α2log⁡W1/W2Γm-N1+1-∫0V∗Vm-N1e-VdVΓm-N1-∫0V∗Vm-N1-1e-VdV.About the integrals ∫0V∗Vm-N1e-VdV and ∫0V∗Vm-N1-1e-VdV, the compound trapezoidal integral formula can be used to obtain integral values. The specific steps are as follows: the interval [0,V∗] is divided into n equal subintervals and each length of subinterval is h=V∗/n. Then, from the compound trapezoidal integral formula, we get (13)∫0V∗gVdV≈h2g0+2∑k=1n-1gV∗kn+gV∗,∫0V∗fVdV≈h2f0+2∑k=1n-1fV∗kn+fV∗,where g(V)=Vm-N1e-V and f(V)=Vm-N1-1e-V. The above two integration results are substituted into (12), and the objective Bayesian estimates of the parameters λ can be obtained.


## 3.2. Bayesian Estimates

With historical data or experts’ experience available, the informative prior could be a better option. Following Doostparast et al. [23], a convenient prior family of distributions for θj(j=1,2) is provided by the power function distributions with probability density function(14)πsθj=βjθjβj-1dj-βj,0<θj<dj;dj,βj>0.We consider the prior density function ofλ as follows:(15)πsλ=Aλa-1exp⁡-bλ,a,b>0,λ>1,where A is the regularization factor.Then the marginal posterior density functions of the parametersλ, θ1, and θ2 are, respectively,(16)πsλ∣t∝λm-N1+a-1exp⁡-α1+α2ln⁡W1-ln⁡W2+bλ,a,b>0,λ>1,πsθj∣t=αjn+βjθjαjn+βj-1dj∗αjn+βj,0<θj<dj∗,dj∗=min⁡tj1,dj,where tj1 is the first-failure time caused by the jth failure cause (j=1,2).Under the squared loss function, the Bayesian estimate ofθj (j=1,2) can be easily obtained as(17)θ^jBS=∫0dj∗θjπSθj∣tdθj=αjn+βjdj∗αjn+βj+1.To obtain the Bayesian estimate ofλ, we expand H(λ)=λm-N1+a and h(λ)=λm-N1+a-1 in Taylor series around the maximum likelihood estimate λ^ to get the following approximation:(18)Hλ≈Hλ^+H′λ^λ-λ^,hλ≈hλ^+h′λ^λ-λ^,where H′(λ)=m-N1+aλm-N1+a-1 and h′λ=m-N1+a-1λm-N1+a-2.Therefore, under the squared loss function, the Bayesian estimate ofλ can be approximated as (19)λ^BS=Hλ^+H′λ^1+I1-1-λ^hλ^+h′λ^1+I1-1-λ^,where I1=(α1+α2)(ln⁡W1-ln⁡W2)+b.


## 3.3. E-Bayesian Estimates

By assuming that the hyperparameters follow certain distributions, the Bayesian estimates can be obtained when the hyperparameters in prior distributions are unknown. However, complicated integration is often a hard work. Han [16, 17] gave the definition of E-Bayesian estimation to overcome this problem. So we will derive the E-Bayesian estimates of the unknown parameters and acceleration factor in this subsection.Suppose that the hyperparametersa and b and βj and dj  (j=1,2) have the joint prior density functions πe(a,b) and πe(βj,dj), respectively. Then the E-Bayesian estimates of λ and θj  (j=1,2) are(20)λ^EB=∬Dλ^BSπea,bdadb,(21)θ^jEB=∬Dθ^jBSπeβj,djdβjddj,where λ^BS and θ^jBS are the Bayesian estimates of λ and θj  (j=1,2), respectively.According to Han [16], a and b and βj and dj  (j=1,2) should be selected to guarantee that πs(λ) and πs(θj) are decreasing functions of λ and θj, respectively. The derivatives of πs(λ) with respect to λ and πs(θj) with respect to θj are (22)∂πsλ∂λ∝λa-2exp⁡-bλa-1-bλ,∂πsθj∂θj=βjβj-1θjβj-2dj-βj,When 0<a<1, b>0, ∂πs(λ)/∂λ<0, and when 0<βj<1, dj>0, ∂πs(θj)/∂θj<0. Thus, πs(λ) and πs(θj) are decreasing functions of λ and θj, respectively.Given that0<a<1 and 0<βj<1, the larger b and dj are, the thinner the tail of πs(λ) and πs(θj) will be. Berger [24] showed that the thinner tailed prior distribution often reduces the robustness of the Bayesian estimate. Consequently, b and dj  (j=1,2) should be given upper bounds l and hj′, respectively. We consider the following two joint prior distributions of a and b and βj and dj  (j=1,2), respectively:(23)π1a,b=2bl2,0<a<1,0<b<l,π2a,b=1l,0<a<1,0<b<l,(24)π1jβj,dj=2djhj′2,0<βj<1,0<dj<hj′,π2jβj,dj=1hj′,0<βj<1,0<dj<hj′.From (19), (20), and (23), we derive the E-Bayesian estimates of λ as(25)λ^EB1=2l2∫0l∫01bHλ^+H′λ^1+I2-1-λ^hλ^+h′λ^1+I2-1-λ^dadb=2λ^l2∫0lb∫01I1+bλ^+m-N1+a1+I1+b1-λ^I1+bλ^+m-N1+a-11+I1+b1-λ^dadb=λ^+λ^l2Q1-Q2,λ^EB2=1l∫0l∫01Hλ^+H′λ^1+I2-1-λ^hλ^+h′λ^1+I2-1-λ^dadb=λ^l∫0l∫01I1+bλ^+m-N1+a1+I1+b1-λ^I1+bλ^+m-N1+a-11+I1+b1-λ^dadb=λ^+λ^lE1-E2,where(26)Q1=l2-I1+N2zλ^2ln⁡I1+lzλ^+N2+I1+N2zλ^2ln⁡I1zλ^+N2+N2lzλ^,Q2=l2-I1+N2-1pλ⌢2ln⁡I1+lpλ⌢+N2-1+I1+N2-1pλ⌢2ln⁡I1pλ⌢+N2-1+N2-1lpλ⌢,E1=I1+l+N2zλ^ln⁡I1+lzλ^+N2-I1+N2zλ^ln⁡I1zλ^+N2,E2=I1+l+N2-1pλ^ln⁡I1+lpλ^+N2-1-I1+N2-1pλ^ln⁡I1pλ^+N2-1,zλ^=λ^+N21-λ^,pλ^=λ^+N2-11-λ^,N2=m-N1.Similarly, from (17), (21), and (24), the E-Bayesian estimate of θj  (j=1,2) can be obtained as (27)θ^jEB1=2hj′31-ln⁡αjn+2αjn+1,dj∗=dj,dj∗1-ln⁡αjn+2αjn+1,dj∗≠dj,θ^jEB2=hj′21-ln⁡αjn+2αjn+1,dj∗=dj,dj∗1-ln⁡αjn+2αjn+1,dj∗≠dj.


## 4. Application of the Pareto Distribution

In this section, we analyze two real data sets to illustrate the application of the Pareto distribution in engineering practice. The first data set (Set I) represent the failure time of 20 mechanical components. These data are taken from Murthy et al. [25] and have been used by Bourguignon et al. [22]. These data are 0.067, 0.068, 0.076, 0.081, 0.084, 0.085, 0.085, 0.086, 0.089, 0.098, 0.098, 0.114, 0.114, 0.115, 0.121, 0.125, 0.131, 0.149, 0.160, and 0.485. The second data set (Set II) is given by Nelson [26], which represents the failure times to breakdown of a type of electronic insulating material subject to a constant-voltage stress. These data are 0.35, 0.59, 0.96, 0.99, 1.69, 1.97, 2.07, 2.58, 2.71, 2.90, 3.67, 3.99, 5.35, 13.77, and 25.50.We check the validity for the Pareto distributionP(α,θ) by using the Kolmogorov test. The statistics of the Kolmogorov test are Dn=supx⁡F(x)-Fn(x)=max1≤i≤20⁡F(xi)-Fn(xi), where F(x) is the distribution function of P(α,θ) and F(x)=1-(θ/x)α, x>0, and Fn(x) is the empirical distribution functions constructed based on the above data Set I and Set II.For the data Set I, when the parametersα=2.113 and θ=0.067, the observation value of the statistic is D^n=max1≤i≤20⁡F(xi)-Fn(xi)=0.1303. For n=20, we choose the value of confidence level α as α=0.01,0.02,0.05, respectively. By using the table of critical value on the Kolmogorov test, we get the critical value D^n,α, where D^20,0.05=0.29403, D^20,0.02=0.32266, and D^20,0.01=0.35241. Since D^n<D^20,0.05, D^n<D^20,0.02, and D^n<D^20,0.01, thus, the fit of Pareto distribution P(α,θ)=P(2.113,0.067) to the above data Set I is reasonable.For data Set II, when the parametersα=0.51 and θ=0.35, the observation value of the statistic is D^n=max1≤i≤20⁡F(xi)-Fn(xi)=0.2187. For n=15, we choose the value of confidence level α as α=0.01,0.02,0.05, respectively. By using the table of critical value on the Kolmogorov test, we get the critical value D^n,α, where D^15,0.05=0.33760, D^15,0.02=0.37713, and D^15,0.01=0.40120. Since D^n<D^15,0.05, D^n<D^15,0.02, and D^n<D^15,0.01, the fit of Pareto distribution Pα,θ=P0.51,0.35 to the above data Set II is reasonable. Therefore, the use of the Pareto distribution is well justified.


## 5. Numerical Example

In this section, we present numerical example using Monte Carlo simulations to illustrate the estimation procedure presented in this paper. The performance of three types of Bayesian estimates for different sample sizes and different progressive censoring schemes (PCS) is assessed based on simulation data. The term “different PCS” means different sets ofRi’s. In this paper, we consider the following three different PCS.Scheme 1.
R 1 = ⋯ = R m - 1 = 0, and Rm=n-m.Scheme 2.
R 1 = R 2 = ⋯ = R L = 0, RL+1=⋯=Rm-1=1, and Rm=n-2m+1+L.Scheme 3.
R 1 = R 2 = ⋯ = R L + 1 = 1, RL+2=⋯=Rm-1=0, and Rm=n-m-1-L, where L=[m/4]+1, [m/4] representing the maximum integer that does not exceed m/4, m≥10.Before progressing further, we first describe how to generate Type II progressively censored competing failure data from Pareto distribution under SSPALT.Step 1.
Givenm, n, and a certain censoring scheme, generate two samples U1,U2,…,Um and W1,W2,…,Wm from the uniform (0,1) distribution [27].Step 2.
Givenαj,θj  (j=1,2), compute t11i and t12i by solving equations Ui=1-θ1α1t11i-α1 and Wi=1-θ2α2t12i-α2, respectively. Then record the minimum of (t11i,t12i) as ti∗ and the corresponding index of the minimum (if t11i<t12i, set ci∗=1; else set ci∗=2) for 1≤i≤m.Step 3.
For prefixed stress changing timeτ, find N1 such that tN1∗≤τ<tN1+1∗. Then let ti=ti∗ and ci=ci∗ for 1≤i≤N1.Step 4.
ForN1+1≤i≤m, generate t21i by solving the equation Ui=1-θ1α1τα1(λ-1)t21i-α1λ, and generate t22i by solving Wi=1-θ2α2τα2(λ-1)t22i-α2λ. Again, take the minimum of (t21i,t22i) as ti∗ as well as the corresponding index of the minimum (if t21i<t22i, set ci∗=1; else set ci∗=2).Step 5.
ForN1+1≤i≤m, let ti=ti∗ and ci=ci∗.
Then{(t1,c1,R1),…,(tm,cm,Rm)} is the required sample.We assumed that the occurrence time of thejth (j=1,2) failure cause of tested product follows Pareto distributions P(α,θ)=P(2.113,0.067) and P(α,θ)=P(1.82,0.051) under use stress levels S0, and λ=2.0 and τ=0.11, the parameters of prior density are taken as a=0.8, b=2.0, β1=0.85, β2=0.6, d1=1.8, and d2=2.1. Considering different m, n and censoring schemes, we generate samples according to the above steps. We replicate the process of evaluating 1000 times in each case and compute the average relative errors (AREs) and mean squared errors (MSEs). ARE and MSE of the estimation θ^ of the parameter θ can be calculated according to the following formulas:(28)AREθ^=11000∑k=11000θ-θ^kθ,MSEθ^=11000∑k=11000θ-θ^k2,where θ^(k) is the kth estimator for the parameter θ.We calculate the AREs and MSEs of the objective Bayesian estimates (OBE), Bayesian estimates (BE), and E-Bayesian estimates (E-BE), respectively. The simulation results are presented in Tables1–3. In Tables 1–3, we denote the E-BE based on the first prior as E-BE1 and the E-BE based on the second prior as E-BE2.Table 1
The AREs and MSEs of all estimates for differentm,n and for censoring Scheme 1.
(n,m) Para. OBEARE  (MSE) BEARE  (MSE) E-BE1ARE  (MSE) E-BE2ARE  (MSE) ( 40,10 ) λ 0.0526 (0.0073) 0.0570 (0.0081) 0.0519 (0.0052) 0.0498 (0.050) θ 1 0.0360 (0.0048) 0.0378 (0.0076) 0.0392 (0.0075) 0.0382 (0.0065) θ 2 0.0474 (0.0115) 0.0485 (0.0138) 0.0512 (0.0128) 0.0508 (0.0107)  ( 40,15 ) λ 0.0514 (0.0069) 0.0551 (0.0067) 0.0515 (0.0048) 0.0495 (0.0044) θ 1 0.0353 (0.0042) 0.0364 (0.0065) 0.0387 (0.0062) 0.0374 (0.0061) θ 2 0.0462 (0.0107) 0.0472 (0.0120) 0.0504 (0.0112) 0.0500 (0.0099)  ( 40,20 ) λ 0.0489 (0.0057) 0.0548 (0.0059) 0.0505 (0.0044) 0.0495 (0.0041) θ 1 0.0336 (0.0032) 0.0341 (0.0056) 0.0372 (0.0051) 0.0365 (0.0049) θ 2 0.0458 (0.0104) 0.0463 (0.0113) 0.0500 (0.0110) 0.0491 (0.0095)  ( 45,10 ) λ 0.0513 (0.0070) 0.0564 (0.0080) 0.0514 (0.0050) 0.0498 (0.0048) θ 1 0.0357 (0.0046) 0.0375 (0.0074) 0.0390 (0.0071) 0.0383 (0.0063) θ 2 0.0471 (0.0111) 0.0486 (0.0127) 0.0505 (0.0115) 0.0512 (0.0100)  ( 45,15 ) λ 0.0510 (0.0064) 0.0549 (0.0058) 0.0514 (0.0047) 0.0491 (0.0046) θ 1 0.0350 (0.0040) 0.0362 (0.0061) 0.0386 (0.0059) 0.0370 (0.0060) θ 2 0.0460 (0.0101) 0.0470 (0.0115) 0.0501 (0.0110) 0.0500 (0.0098)  ( 45,20 ) λ 0.0484 (0.0056) 0.0542 (0.0054) 0.0504 (0.0041) 0.0493 (0.0038) θ 1 0.0334 (0.0030) 0.0338 (0.0051) 0.0370 (0.0048) 0.0362 (0.0048) θ 2 0.0454 (0.0100) 0.0462 (0.0111) 0.0497 (0.0105) 0.0489 (0.0087)  ( 50,10 ) λ 0.0511 (0.0068) 0.0563 (0.0076) 0.0511 (0.0047) 0.0494 (0.0046) θ 1 0.0353 (0.0044) 0.0373 (0.0071) 0.0387 (0.0070) 0.0380 (0.0061) θ 2 0.0468 (0.0109) 0.0483 (0.0118) 0.0502 (0.0112) 0.0510 (0.0094)  ( 50,15 ) λ 0.0509 (0.0062) 0.0546 (0.0058) 0.0512 (0.0045) 0.0492 (0.0045) θ 1 0.0348 (0.0037) 0.0359 (0.0060) 0.0382 (0.0058) 0.0368 (0.0058) θ 2 0.0459 (0.0095) 0.0468 (0.0110) 0.0500 (0.0107) 0.0500 (0.0093)  ( 50,20 ) λ 0.0481 (0.0054) 0.0541 (0.0051) 0.0503 (0.0040) 0.0491 (0.0035) θ 1 0.0330 (0.0027) 0.0335 (0.0047) 0.0368 (0.0042) 0.0369 (0.0043) θ 2 0.0445 (0.0096) 0.0462 (0.0106) 0.0497 (0.0102) 0.0488 (0.0072)Table 2
The AREs and MSEs of all estimates for differentm,n and for censoring Scheme 2.
( n , m ) Para. OBEARE (MSE) BEARE (MSE) E-BE1ARE (MSE) E-BE2ARE (MSE) ( 40,10 ) λ 0.0531 (0.0080) 0.0576 (0.0084) 0.0521 (0.0055) 0.0501 (0.0053) θ 1 0.0365 (0.0051) 0.0381 (0.0081) 0.0398 (0.0079) 0.0386 (0.0067) θ 2 0.0478 (0.0118) 0.0488 (0.0142) 0.0511 (0.0125) 0.0516 (0.0117)  ( 40,15 ) λ 0.0516 (0.0071) 0.0551 (0.0082) 0.0515 (0.0051) 0.0495 (0.0049) θ 1 0.0357 (0.0046) 0.0364 (0.0080) 0.0387 (0.0064) 0.0374 (0.0061) θ 2 0.0465 (0.0113) 0.0472 (0.0138) 0.0504 (0.0114) 0.0500 (0.0112)  ( 40,20 ) λ 0.0489 (0.0077) 0.0548 (0.0078) 0.0505 (0.0058) 0.0495 (0.0056) θ 1 0.0336 (0.0044) 0.0341 (0.0075) 0.0372 (0.0056) 0.0365 (0.0053) θ 2 0.0458 (0.0110) 0.0463 (0.0133) 0.0500 (0.0112) 0.0491 (0.0110)  ( 45,10 ) λ 0.0527 (0.0079) 0.0564 (0.0079) 0.0511 (0.0052) 0.0501 (0.0051) θ 1 0.0365 (0.0048) 0.0380 (0.0077) 0.0394 (0.0075) 0.0382 (0.0065) θ 2 0.0474 (0.0115) 0.0485 (0.0133) 0.0509 (0.0122) 0.0512 (0.0114)  ( 45,15 ) λ 0.0513 (0.0067) 0.0549 (0.0075) 0.0513 (0.0050) 0.0492 (0.0048) θ 1 0.0348 (0.0037) 0.0364 (0.0074) 0.0382 (0.0062) 0.0370 (0.0058) θ 2 0.0464 (0.0112) 0.0470 (0.0131) 0.0501 (0.0108) 0.0487 (0.0107)  ( 45,20 ) λ 0.0486 (0.0065) 0.0546 (0.0071) 0.0501 (0.0057) 0.0492 (0.0045) θ 1 0.0332 (0.0034) 0.0341 (0.0070) 0.0372 (0.0056) 0.0365 (0.0051) θ 2 0.0451 (0.0110) 0.0457 (0.0127) 0.0492 (0.0104) 0.0490 (0.0104)  ( 50,10 ) λ 0.0521 (0.0075) 0.0560 (0.0076) 0.0508 (0.0046) 0.0500 (0.0043) θ 1 0.0362 (0.0046) 0.0380 (0.0075) 0.0392 (0.0074) 0.0381 (0.0063) θ 2 0.0471 (0.0113) 0.0483 (0.0131) 0.0504 (0.0120) 0.0510 (0.0112)  ( 50,15 ) λ 0.0504 (0.0065) 0.0546 (0.0074) 0.0512 (0.0050) 0.0491 (0.0045) θ 1 0.0354 (0.0036) 0.0364 (0.0072) 0.0382 (0.0061) 0.0370 (0.0055) θ 2 0.0464 (0.0110) 0.0470 (0.0130) 0.0501 (0.0105) 0.0496 (0.0104)  ( 50,20 ) λ 0.0475 (0.0059) 0.0536 (0.0072) 0.0491 (0.0048) 0.0487 (0.0044) θ 1 0.0324 (0.0029) 0.0332 (0.0170) 0.0366 (0.0052) 0.0356 (0.0051) θ 2 0.0443 (0.0107) 0.0450 (0.0127) 0.0483 (0.0102) 0.0481 (0.0100)Table 3
The AREs and MSEs of all estimates for differentm,n and for censoring Scheme 3.
( n , m ) Para. OBEARE (MSE) BEARE (MSE) E-BE1ARE (MSE) E-BE2ARE (MSE) ( 40,10 ) λ 0.0535 (0.0079) 0.0578 (0.0082) 0.0521 (0.0056) 0.0510 (0.0054) θ 1 0.0358 (0.0053) 0.0385 (0.0083) 0.0399 (0.0082) 0.0390 (0.0072) θ 2 0.0481 (0.0129) 0.0491 (0.0145) 0.0512 (0.0130) 0.0520 (0.0122)  ( 40,15 ) λ 0.0523 (0.0072) 0.0551 (0.0080) 0.0520 (0.0052) 0.0501 (0.0051) θ 1 0.0364 (0.0048) 0.0371 (0.0077) 0.0396 (0.0076) 0.0380 (0.0071) θ 2 0.0471 (0.0122) 0.0480 (0.0142) 0.0511 (0.0129) 0.0512 (0.0120)  ( 40,20 ) λ 0.0491 (0.0066) 0.0552 (0.0078) 0.0513 (0.0051) 0.0504 (0.0050) θ 1 0.0342 (0.0043) 0.0350 (0.0075) 0.0381 (0.0071) 0.0373 (0.0068) θ 2 0.0465 (0.0120) 0.0469 (0.0140) 0.0507 (0.0124) 0.0499 (0.0118)  ( 45,10 ) λ 0.0532 (0.0075) 0.0571 (0.0079) 0.0521 (0.0052) 0.0504 (0.0049) θ 1 0.0363 (0.0051) 0.0381 (0.0081) 0.0398 (0.0081) 0.0390 (0.0067) θ 2 0.0480 (0.0127) 0.0491 (0.0141) 0.0512 (0.0128) 0.0520 (0.0120)  ( 45,15 ) λ 0.0518 (0.0068) 0.0551 (0.0075) 0.0520 (0.0050) 0.0501 (0.0046) θ 1 0.0358 (0.0050) 0.0371 (0.0075) 0.0391 (0.0076) 0.0381 (0.0064) θ 2 0.0467 (0.0118) 0.0478 (0.0137) 0.0510 (0.0123) 0.0510 (0.0117)  ( 45,20 ) λ 0.0491 (0.0060) 0.0552 (0.0072) 0.0513 (0.0047) 0.0450 (0.0042) θ 1 0.0340 (0.0048) 0.0345 (0.0069) 0.0381 (0.0065) 0.0371 (0.0062) θ 2 0.0462 (0.0115) 0.0470 (0.0129) 0.0503 (0.0120) 0.0493 (0.0113)  ( 50,10 ) λ 0.0530 (0.0074) 0.0571 (0.0067) 0.0520 (0.0050) 0.0502 (0.0045) θ 1 0.0353 (0.0050) 0.0373 (0.0078) 0.0387 (0.0079) 0.0380 (0.0064) θ 2 0.0468 (0.0124) 0.0483 (0.0140) 0.0502 (0.0124) 0.0510 (0.0119)  ( 50,15 ) λ 0.0510 (0.0067) 0.0546 (0.0064) 0.0512 (0.0056) 0.0492 (0.0051) θ 1 0.0348 (0.0042) 0.0359 (0.0075) 0.0382 (0.0076) 0.0368 (0.0062) θ 2 0.0459 (0.0120) 0.0468 (0.0137) 0.0500 (0.0121) 0.0500 (0.0115)  ( 50,20 ) λ 0.0481 (0.0065) 0.0541 (0.0062) 0.0503 (0.0053) 0.0491 (0.0050) θ 1 0.0330 (0.0040) 0.0335 (0.0072) 0.0368 (0.0073) 0.0369 (0.0058) θ 2 0.0454 (0.0116) 0.0462 (0.0134) 0.0497 (0.0118) 0.0485 (0.0113)From Tables1–3, the following observations can be made:(1)
For fixedn, asm increases the AREs and MSEs decrease for all estimates.(2)
In terms of the values of AREs and MSEs, the OBE performances are better than that of BE, and E-BE2 performances are better than that of E-BE1 under the different progressive censoring schemes.(3)
Under the different progressive censoring schemes, for all estimates, the values of AREs are less than 0.06, and the values of MSEs are less than 0.015. Hence the results of all estimates are satisfied.


## 6. Conclusions

In this paper, the Bayesian analysis for the step-stress partially accelerated competing failure model from Pareto distribution is investigated under Type II progressive censoring. Based on the noninformative prior and informative prior distributions, we derive the objective Bayesian estimates and Bayesian estimates of the unknown parameters and acceleration factor, respectively. Considering such case that the hyperparameters are unknown and follow certain distributions, the E-Bayesian method is utilized to estimate the unknown parameters and acceleration factor. We analyze two real data sets to illustrate the application of the Pareto distribution in engineering practice. Finally, the comparisons of the efficiencies of the objective Bayesian estimates, Bayesian estimates, and corresponding E-Bayesian estimates are conducted. It is concluded that the OBE performances are better than that of BE and E-BE2 performances are better than that of E-BE1 under the different progressive censoring schemes. In addition, we discuss the effects of sample sizes and different progressive censoring schemes on the estimates. The simulation results demonstrate that for fixedn the AREs and MSEs of all estimates decrease with m increasing.



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