A Nonconstant Shape Parameter-Dependent Competing Risks’ Model in Accelerate Life Test Based on Adaptive Type-II Progressive Hybrid Censoring
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.
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.  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.  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  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.  discussed the Bayes estimations for Marshall–Olkin bivariate exponential (MOBE) distribution. Shen and Xu  analyzed the statistical inference of dependent competing risks’ model using the Marshall–Olkin bivariate Weibull (MOBW) distribution. Bai et al.  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  discussed the Bayes estimations (BEs) of the MOBW distribution. Feizjavadian and Hashemi  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. , 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 is fixed in advance, and the experiment time is allowed to run over the prefixed time . The progressive censoring scheme is allowed to change according to the m failure times . For convenience, let . At the time of the first failure, units are randomly removed from the remaining units. Similarly, at the second failure , units from the remaining units are randomly removed and so on. If the mth observed failure occurs before time (i.e., ), the test stops at time , and all remaining units are removed, where . Otherwise, if , where , and we adapt the number of units removed from the test upon failures by setting and . In this case, the actually applied progressive censored scheme is . Presently, some scholars discussed the AT-II PHCS, i.e., Cramer and Iliopoulos  derived some distributional results and showed that maximum likelihood estimators coincide with those in deterministic progressive type-II censoring. Hemmati and Khorram  discussed the AT-II PHCS in the presence of the competing risk. Nassar and Abo-Kasem  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  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 , Perozek , and Willemse and Koppelaar  adopted this distribution to describe the adult lifespan by demographers and actuaries. Finch and Pike  and Ricklefs and Scheuerlein  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  considered the BEs of the Gompertz distribution in the competing risks’ model under progressively hybrid censoring. Liu et al.  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.  investigated the MLEs and BEs of the Gompertz distribution based on the generalized progressively hybrid censored scheme. Dey et al.  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 Section 2, 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 parameters and can be represented as follows, respectively:
The reliability function of Gompertz distribution can be expressed as follows:where is the shape parameter and is the scale parameter. We denote the Gompertz distribution with parameters and as .
Suppose that the random variable are independent of each other and . Let and , then the bivariate random variable is followed by the MOBG distribution with shape parameter and scale parameters , and , and it is denoted by . Thus, the joint survival function of is shown as follows:
And, the joint PDF of can be obtained as the following expression:where
2.2. Model Description
In this section, the dependent competing risks’ model in CSALT under AT-II PHCS is considered.
Let be the accelerated stress levels and be the normal stress level. The AT-II PHCS under CSALT can be run as follows. Under the ith stress level , and suppose identical units are put on the life test, the observed failure number , terminal time , and progressive censoring scheme are given in advance. For the given , if , where , the removed units from the test upon failures is , and . Otherwise, .
The following basic assumptions are made for the statistical inference of the dependent competing risks’ model. A1: under the stress level ; for the jth unit, there are two random variables and , which are the failure time under first and second failure causes, respectively. Thus, the failure time of the unit is . A2: under the stress level ; the pair of random variable follows a . follows , follows , and follows where, , , and . A3: the accelerated function of the scale parameter is log-linear:
The accelerated function of the shape parameter is log-linear:where , and are the accelerated factors and the unknown parameters and is the known function of the stress level
Based on the model description, we can obtain the adaptive type-II progressive hybrid censored competing risks’ data , where is failure cause of the ith unit, and it is defined as follows:while indicates that the failure cause is uncertain.
Under the stress level , let represent the failure numbers of those caused by both risks, first risk and second risk, where is the indicator function; . In addition, represents the numbers of the failure units which have the failure time, while not having the corresponding failure cause. Let and , and we can obtain the following result.
Theorem 1. Under the stress level ,where .
Proof. Under the stress level , each unit failure is caused by one of , , or . Probability of each case isFor the given , and , we can obtain byThen, the theorem is completed.
3. Point Estimations
3.1. Maximum Likelihood Estimations
Let is the censored sample in CSALT and is the sample observation. Let ; under the stress level , based on the dependent competing risks’ data, the likelihood function can be expressed as follows:where
So, we can obtainwhere .
The full likelihood function is
Based on equation (16), the log-likelihood function can be transformed into the following result:
The MLEs of unknown parameters can be obtained by the maximum function in equation (17). By setting the first partial derivative of with respect to to zero, we can obtain following results:
By solving equation (18), we can get the MLE of , as the following form:
Substituting into equation (19), we can obtain the equation for :
Theorem 2. Based on the censored data and the progressive scheme , the MLEs and of the parameters and exist and are unique if and only if
Proof. From equations (20) and (21), we can obtain that and exist and are unique by showing that exists and is unique if and only if condition (22) holds. The derivative of can be expressed as follows:Let , we havewhereSince for all , we have .
Now, letThen, we haveBy using the Cauchy–Schwarz inequality, we can obtain . Then, it follows that which implies that is a decrease function. Therefore, equation (21) has the unique root if and only if and :If and only if equation (22) is satisfied, :Then, the theorem is completed.
Theorem 3. Using the MLE of the parameter , we can get the MLE of the parameter having the following properties:(1) is the asymptotic unbiased estimator of (2) is a consistent estimator of (3) is a sufficient statistics of
Proof. From A2, follows the Gompertz distribution with the shape parameter and scale parameter . By the variable transformed method, the random variable follows the exponential distribution with mean .
LetWe can obtain that are independent and identical distribution as the exponential distribution with mean . According to the relationship between the exponential distribution and gamma distribution, we haveLet , then :Then, follows the inverse gamma distribution with the parameters and , and it is denoted by .
Using equation (20),Then, we haveThe proof of statement 1 is completed.
Using equation (20), we can obtainThen,The proof of statement 2 is completed.
Finally, under the stress level , the likelihood function can be written as follows:Based on equation (20), we can obtainThen, the likelihood function can be written as follows:The likelihood function can be decomposed into two functions and , 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 parameters , and are independent of each other. According to Wu and Shi , for the parameter , we select a finite number of values on the interval , which is denoted by . The PDF of the parameters is , where .
Based on equation (16), we can obtain the conditional probability density function of parameter :
When , the conditional probability density function of parameter can be expressed as follows:
Then, we can obtain that belongs to the gamma distribution family.
Under the condition , the condition prior distribution of the parameter , is , which is the conjugate distribution for the parameter . The PDF can be expressed as follows:
The condition posterior density function of the parameter is
Using the binomial expansion,
The likelihood function in equation (15) can be written as the following form:
Then, the condition posterior density function of the parameter can be obtained as follows:where .
The marginal posterior density distribution of the parameter can be obtained as follows:where
Under the squared error loss function, the BEs of , and can be written as follows:
3.3. The Estimation of Reliability
The reliability estimation of the unit at any time t under the normal stress level is
The estimation of the shape parameter under normal stress level can be obtained from the equation . And, the estimation of the scale parameter under normal stress level can be obtained from the equation . Putting the estimations of the parameters into equations (5) and (6), we can obtain the estimations of by the least square method:
4. Interval Estimation
4.1. Approximation Confidence Intervals
For the parameters and , 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 parameters and