Parameters Estimation of the 3-Component Mixture Models of Exponential Family under Bayesian Paradigm

Abstract

This thesis is concerned with the problem of estimating the parameters of the 3-component mixture models of the members of one parameter Exponential family using type-I right censored data under Bayesian paradigm. The models include: (i) Exponential distribution (ii) Rayleigh distribution (iii) Pareto distribution and (iv) Burr Type-XII distribution. In the Bayesian perspective, these 3-component mixtures of distributions get either no or least consideration in literature so far. Also, the reliability analysis of the 3-component mixtures of distributions is presented in this study. The expressions for the Bayes estimators and their posterior risks using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function are derived. The censored sampling environment is considered due to its popularity in reliability theory and survival analysis. When prior information is available, elicitation of hyperparameters through prior predictive method based on predictive probabilities is given. The posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. In addition, the limiting expressions for the Bayes estimators and posterior risks under different loss functions are derived. To examine, numerically, the performance of the Bayes estimators under different loss functions, we have simulated Bayes estimates and posterior risks for different sample sizes, test termination times and parametric values. Further, to highlight the practical significance of each 3- component mixture, the Bayesian analysis of the real life mixture data sets is conducted. At the end of this thesis, the conclusion is presented. Some recommendations for future research are also given.