Bayesian Statistical Analysis For Masked Data

Competing cause of failure is an important cause of failure of product. However, due to many reasons, such as lack of proper diagnostic equipment, cost and time constraints associated with failure analysis, recording errors, the destructive nature of certain failed components that make exact diagnosis impossible and so on, it turns out that causes of failure are usually masked in practice. Many authors have proposed lots of approaches to analyze masked data. However, all of them are based on the following three assumptions: 1. The components' lifetime are independent of each other, and the distributions of which are known; 2. Life test is under use condition; 3. The systems considered are all series systems. Expect for relaxing these assumptions by using Bayesian method, we also consider some problems associated with masked data. The contents of this dissertation are as follows:In Chapter two, we present Bayesian analysis for masked data when the lifetime of the components follow Pareto distributions. For the case of the series system with two components, we utilize two kinds of priors for the scale parameters in the model, and obtain the posterior estimates by using Gibbs sampling. In the simulation, we show that the posterior estimates of parameters are affected by the information loss of masked data and choice of hyperparameters. Then, for the case of the series system with J> 2 components, we give the Gibbs sampling procedure, and propose several approaches to analyze high degree censored data in the real data analysis.In Chapter three, we consider the nonparametric Bayesian analysis of masked data, and propose two kinds of nonparametric Bayesian methods. Firstly, for relaxing the as-sumption that the components are independent of each other, we choose Dirichlet process as the prior for system's survival function. Then we obtain the posterior distribution of the survival function, and show the consistence of the posterior estimate. Second, assume that the components are independent of each other, the multivariate Dirichlet process is used as the prior for the subsurvival functions, based on which the posterior distribution is obtained. Then the posterior distributions of the components'survival function and the system's survival function are obtained by using Peterson's formula.In Chapter four, we study the objective Bayesian analysis of masked data. Firstly, we briefly introduce three most used noninformative priors in the objective Bayesian analysis, and specify the difference and connection among these three priors by an example. Then we study three kinds of problems. In the first part, we consider the objective Bayesian analysis of masked data under symmetric assumption. We show the Bayesian estimates base on a noninformative prior coincide with maximum likelihood estimates. However, the posterior distribution is improper when the sample of size is small. Hence the Jeffreys prior and the reference prior are derived, based on which the posterior distributions are always proper. Then the frequentist coverage probability of the a-quantile of the posterior distributions is derived. In the seconde part, the objective Bayesian analysis of two-stage masked data is considered. Four noninformative priors are derived, and the Bayesian estimates based on these priors are compared with maximum likelihood estimates. Finally, we consider the overparameterized models that exist in most statistical analysis. We give the reference priors of the estimable parameters in these models, as well as the properties of the reference priors. For illustration, three examples are studied, especially, a solution of unestimable problem in the masked data analysis is given in the third example.In Chapter five, we study the objective Bayesian analysis of accelerated competing failure models under type-I censoring. A two-step estimation procedure is given when the lifetime of components follow Weibull distributions. In the first step, for the param-eters of the distribution of components'lifetime at each stress level, two noninformative priors (Jeffreys prior and reference prior) are derived and the posterior propriety of the parameters is shown based on these two priors. Then a specified Gibbs sampling proce-dure is given to obtain the posterior estimates of the parameters. When the conditional posterior distributions of the parameters are improper, we propose a method to modify the likelihood function so that the proper posterior distributions of the parameter can be obtained. In the second step, the least squares approach is used to obtain the estimates of the parameters in the accelerated function. Finally, we give a simulation and two real examples to show the effectiveness of the methodIn Chapter six, we consider the Bayesian analysis of masked data in step-stress ac-celerated life testing. In step-stress ALT, the observed lifetimes are not the exact system lifetimes, which need time-shift based on Nelson's (1980) cumulative exposure model. Due to the unknown causes of failure in the masked data, time-shift can not be conducted directly. We assume that the lifetime of the components of the series system are Weibull distributions, and that the accelerated function is log-linear. Bayesian method is pre-sented to overcome the problem of time-shift by using auxiliary variables, and a Gibbs sampling procedure is proposed to obtain the estimates of the parameters. The developed techniques are illustrated by a numerical example.In the last Chapter, we study the hypothesis of masking probability assumption and Bayesian analysis of masked data in the parallel system. For the former one, Bayes factor is used to select masking probability assumption. When the lifetimes of components follow exponential distributions or Pareto distributions, we show the consistency of Bayes factor. For the latter one, we propose a general framework of Bayesian analysis for parametric model, and use two nonparametric Bayesian methods to estimate the reliability of the system in the nonparametric model.