Abstract
The bivariate or multivariate distribution can be used to account for the dependence structure between different failure modes. This paper considers two dependent competing failure modes from Gompertz distribution, and the dependence structure of these two failure modes is handled by the Marshall–Olkin bivariate distribution. We obtain the maximum likelihood estimates (MLEs) based on classical likelihood theory and the associated bootstrap confidence intervals (CIs). The posterior density function based on the conjugate prior and noninformative (Jeffreys and Reference) priors are studied; we obtain the Bayesian estimates in explicit forms and construct the associated highest posterior density (HPD) CIs. The performance of the proposed methods is assessed by numerical illustration.
1. Introduction
It is extremely common that the failure of a product or a system contains several competing failure modes in reliability engineering; any failure mode will lead to the failure result. Competing risks’ data contain the failure time and the corresponding failure mode, which can be modeled by the competing risks’ model and has been commonly performed in many research fields, such as engineering and medical statistics. Previous studies have mostly assumed the competing failure modes to be independent; Wang et al. [1], Ren and Gui [2], and Qin and Gui [3] focused on the independent competing risks’ model under progressively hybrid censoring from Weibull and Burr-XII distributions. Objective Bayesian analysis for the competing risks’ model with Wiener degradation phenomena and catastrophic failures was studied by Guan et al. [4]. In practice, the independency relationship between different failure modes is a very special case; a more common situation is dependency. That is, the failure mechanisms are interactive and interdependent; the occurrence of one failure mode will affect the occurrence of other failure modes. For example, a ship fixed carbon dioxide fire extinguishing system can fail due to pressure gauge, distribution value, cylinder group, and so on; these failure modes are dependent because they all are related to the storage environment. Therefore, it is more reasonable to assume dependency among different competing failure modes. The competing risks’ model considers the product or system with multiple dependent competing failure modes, any one of which will cause the occurrence of failure. The dependent competing risks’ model has been extensively studied. Zhang et al. [5] and Zhang et al. [6] studied the dependent competing risks’ model under accelerated life testing (ALT) by copula function to measure the dependence between different competing failure modes; the results indicate the copula construction method has good accuracy and universality. Wang and Yan [7] and Wu et al. [8] also studied this model under ALT and progressively hybrid-censoring scheme using Clayton copula and Gumbel copula, respectively. For other related works, see the works of Lo and Wike [9] and Fang et al. [10].
In addition to using copula function to handle the relationship between different competing failure modes, the bivariate or multivariate distribution also can be used to account for the correlation between different failure modes. The Marshall–Olkin distribution [11], which has many good properties, is the best-known bivariate distribution and has been discussed extensively; it has a parameter to describe the dependence structure. Li et al. [12], Kundu and Gupta [13], and Bai et al. [14] provided reviews on Marshall–Olkin–Weibull distribution; Kundu and Gupta [13] obtained the explicit forms of the unknown parameters when the shape parameter is known; when the shape parameter is unknown, they used the importance sampling to compute the Bayesian estimates of the unknown parameters. Bai et al. [14] discussed the statistical analysis for the accelerated dependent competing risks’ model under Type-II hybrid censoring schemes. Guan et al. [15] studied objective Bayesian analysis for the Marshall–Olkin exponential distribution based on reference priors; they also found that some of the reference priors are also matching priors and the posterior distributions based on these priors are proper.
The Gompertz distribution is a widely used growth model which has been studied extensively; Ismail [16] studied the Bayesian analysis of Gompertz distribution parameters and acceleration factor in the case of partially accelerated life testing under Type-I censoring scheme. Ghitany et al. [17] considered a progressively censored sample from Gompertz distribution; they discussed the existence and uniqueness of the MLEs of the unknown parameters. The Gompertz distribution plays an important role in fitting clinical trials’ data in medical science and can be used to the theory of extreme-order statistics. In this paper, we will study the dependent competing risks’ model from the Marshall–Olkin bivariate Gompertz (MOGP) distribution, which is a bivariate distribution with Gompertz marginal distributions. We focus our attention on the statistical analysis of the model parameters, including classical likelihood inference, Bayesian analysis, and objective Bayesian analysis. Because the Bayesian analysis based on conjugate prior is sensitive to the hyperparameters, inappropriate choice of it will cause bad priors. Based on this reason, we propose the objective Bayesian analysis based on noninformative priors for comparison. The objective Bayesian inference has been studied by Guan et al. [14], Bernardo [18], and Berger and Bernardo [19] based on Reference and Jeffreys priors.
In the rest of this paper, we will present the model description and some properties. Section 3 presents the MLEs and associated bootstrap CIs. In Section 4, Bayesian estimates and associated HPD CIs based on conjugate prior, Jeffreys prior [20], and reference priors [18] are obtained, and these priors lead to proper posteriors which are proved. Section 5 presents some results obtained from simulation study and illustrative analysis. Section 6 gives some final concluding remarks.
2. Model Description
Suppose that is a Gompertz distribution; the density function and reliability function of it arewhere is shape parameter and is scale parameter.
Suppose are three independent Gompertz variables with different scale parameters, that is, , , and . Let and ; we obtain and . Then, the pair of variables follows the MOGP distribution denoted by . When , the two variables are independent and will be dependent when ; hence, can be regarded as a correlation coefficient between .
The joint PDF of can be written as
The surface plots of are presented in Figure 1. From Figure 1, we can see that is a unimodal function.
(a)
(b)
(c)
(d)
Put n identical products into test, and each product has two dependent failure modes with lifetimes , . Then, the system lifetime is . Let , , and , for , where is an indicator function. Then, we can compute , , and .
Theorem 1. For , , , and , We have
Proof. For , we have ,Therefore, .
The likelihood function iswhereThen, we obtain
3. Classical Inference
3.1. Maximum Likelihood Estimates (MLEs)
The MLEs of can be obtained by maximizing the logarithm of . Set the first partial derivation of about to 0, i.e.,
From (8), we get the MLEs of as
Substituting into , we obtainwhich is the profile logarithm likelihood function of .
We can show that , which implies that is concave. Some iterative schemes can be used to find the MLE for , such as Newton–Raphson algorithm.
3.2. Bootstrap Confidence Intervals (CIs)
Since it is hard to construct the exact CIs for the unknown parameters, we consider the Bootstrap method to construct CIs for parameters . The Bootstrap method is a resampling method to estimate some statistical characteristics for the unknown parameters by taking samples from the original samples repeatedly; the obtained samples are called Bootstrap samples. This method has a great practical value since it does not need to assume the overall distribution or construct the pivot quantity. We generate the Bootstrap sample by the following three steps: Step 1: for the fixed value of n and observed data , we get the estimates based on the maximum likelihood method. Step 2: for the values of n, , we generate the sample . Then, get the MLEs . Step 3: repeat Step 2 M times to obtain M sets of the values . Arrange them as follows to get the Bootstrap sample:
Based on the Bootstrap sample and by percentile Bootstrap (Boot-P) method, we construct the Boot-P CIs for at confidence level as
4. Bayesian Inference and HPD CIs
4.1. Conjugate Prior
In this section, we suppose the shape parameter is known. Denote , which has a Gamma prior with hyperparameters a and b as
Due to , so given , follows a Dirichlet prior with hyper parameters , and , that is,
Therefore, the joint prior of becomeswhere .
4.2. Jeffreys Prior
According to Jeffreys [20], Jeffreys prior is proportional to the square root of the determinant of the Fisher information matrix. From (7), we obtain the Fisher information matrix of as
From Theorem 1, we have , so can be written as
Thus, the Jeffreys prior is given by
Theorem 2. Based on the Jeffreys prior , the joint posterior distribution of is proper.
Proof. From (6) and (7), we obtain the joint posterior distribution of based on asIntegrating with respect to , we obtainwhere and is a beta function.
Thus, the joint posterior distribution of based on is proper.
4.3. Reference Priors
Bernardo [18] and Berger and Bernardo [19] proposed the reference prior which plays a vital role in the objective Bayesian inference. We set , , and ; the transformation from to is one-to-one with the inverse transformation , , and . The Jacobian matrix of the transformation has the form
The likelihood function (3) becomes
The Fisher information matrix of can be written as
Theorem 3. (i)Under the ordering groups and , the reference priors are the same, which is given by ; the corresponding reference prior for is (ii)Under the ordering groups , , , and , the reference priors are the same, which is given by ; the corresponding reference prior for is (iii)Under the ordering groups and , the reference priors are the same, which is given by ; the corresponding reference prior for is
Proof. (i)The Fisher information matrix of is where and . The reference prior for the ordering groups and is the same as in [21], which is given by(ii)The inverse of is(iii)According the notations in [18], we obtain , , and . Choose the compact sets , such that , , and , as . Then, we have where : Then, we get the reference prior as where is an inner point of . Similarly, under the ordering group , the reference prior is . The Fisher information matrix of is The inverse of is Similarly, we obtain , , and . Choose the compact sets , such that , , and , as . Then, we have where , Let be an inner point of ; we get the reference prior as Similarly, under the ordering group , the reference prior is .(v)The Fisher information matrix of isThe inverse of isThen, we obtain , , and .
Choose the compact sets , such that , , and , as . Then, we havewhere ,Let be an inner point of , we obtain the reference prior asSimilarly, under the ordering group , the reference prior is . According to the one-to-one transformation from to , we can obtain the reference priors from , respectively.
Theorem 4. Based on the reference priors and , the posterior distributions of are proper.
Proof. The joint posterior distributions of based on reference prior and are, respectively, asIntegrating and with respect to , respectively, we obtainThus, the posterior distributions of based on and are proper.
4.4. Bayesian Estimates
The joint posterior distributions of based on are, respectively, aswherewhere and .
Thus, we obtain
Similarly,where
We obtainwhere