Objective Bayesian Analysis Of Frechet Distribution And Inference For Reliability Analysis

Bayesian models usually require the specification of prior distributions for param-eters. The choice of a prior distribution can be difficult when there is limited prior knowledge available, so it is useful to have default objective priors which avoid the need for subjective specifications. Present study focuses on the performance of certain noninformative prior distributions in the context of parameter estima-tion. Parameter estimation plays a critical role towards accurately identifying the system behavior through mathematical models. Each method suggested has its own merits, and a method should be chosen based on the features of data being analyzed and goals of the analysis. Consequently, the estimation of parameters from sampled data have been received a significant attention in the literatures. This dissertation consists of three main parts.The purpose of the first part of the research is to estimate two-unknown parame-ters of the Frechet distribution using reference prior. The reference prior is derived and general forms of probability matching priors are also obtained in case of any parameter of interest and concluded that the reference prior is also a second order matching prior. Since the Bayesian estimators cannot be obtained in closed form, they are obtained using Monte Carlo simulation and Laplace approximation. The Bayesian and maximum likelihood estimates are compared via simulation study and the results show that the Bayesian estimators perform better than maximum likelihood estimators even in small sample sizes. As for the case of censored sample, Bayesian estimators are obtained under the assumption of noninformative priors with respect to LINEX (linear-exponential) loss and general entropy loss functions. The Bayesian estimators cannot be obtained in closed form and Lindley’s approx-imation is used to compute the approximate Bayesian estimates, compared with their maximum likelihood counterparts using Monte Carlo simulation.The second part of the study focuses on reliability estimation of stress-strength model in the context of objective Bayesian approach assuming that both stress and strength follow the Frechet distribution with different scale parameters and com-mon shape parameter. Several reference priors and the general form of matching priors are derived. It is concluded that none of the reference priors is a match-ing prior. Further, the propriety of posterior based on these reference priors and matching priors are shown. The study showed that matching prior performs better than Jeffreys prior and reference priors, in terms of target coverage probabilities. In third part, an attempt has been made to analyze and plan a simple step-stress accelerated life test when the lifetimes at each stress level follow the Frechet distri-bution, the scale parameter is a log-linear function of the stress, and a cumulative exposure model are assumed. The point and interval estimates of the model pa-rameters are obtained using classical method and Bayesian method. Further, the sensitivity analysis is carried out in order to investigate how changes in priors will affect the estimate of model parameters. The Bayesian optimal plan is ob-tained by minimizing the pre-posterior variance for the pth percentile of the lifetime distribution at normal stress condition with type-I censoring via Monte Carlo sim-ulation algorithm based on Gibbs sampling. The results indicated that Bayesian approach performs well in analyzing and designing reliability life test when there is prior information and uncertainty for the model parameters.