Some Conditions in Bayesian Analysis for Mixture of the Transmuted Distributions

Abstract

This study presents the Bayesian inference of single and mixtures of some subclasses of transmuted family of distribution. In particular, the focus of the study is on Transmuted Fréchet (TFr), Transmuted Weibull (TW) and Transmuted Pareto (TPa) distributions.The mixtures of these distributions are studied under type−I censoring scheme. As per our knowledge, there is no literature available on Bayesian analysis for the transmuted family of distributions. Different types of Noninformative and Informative priors are assumed for the derivation of posterior distributions. Furthermore, the Bayes estimators are evaluated under three types of loss functions namely; Squared error loss function (SELF), Precautionary loss function (PLF) and Quadratic loss function (QLF). An extensive simulation study is conducted to show some interesting properties and comparisons of the Bayes estimates in terms of sample sizes, censoring rates, mixing proportions and various combinations of the component density parameters. In each case, the Bayes estimators for the parameters of the subject models are found to be intractable and Markov Chain Monte Carlo simulations are used to compute them, numerically. In addition, Bayesian credible intervals (BCIs) are also constructed in this study. An ordinary type−I right censoring scheme is considered throughout the dissertation and Bayes estimators, posterior risks, credible intervals, expected test termination time which is the most important for life testing experiments, are discussed and evaluated. Real data sets are used to illustrate the proposed methodology. By comparing the results of simulation study and real data set; it is worth to be mentioned that the results are compatible and the Bayes estimates based on informative priors produced more efficient estimates than under noninformative priors. Furthermore, SELF is observed more appropriate for the estimation of unknown parameters than PLF and QLF as it has minimum risk.