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Author

Yimin Ma

Date of Award

11-1998

Degree Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Supervisor

Professor N. Balakrishnan

Abstract

Modern Bayesian analysis and empirical Bayesian analysis are dominated by the exponential distribution family; lots of research works have been done in the literature. However, for nonexponential distribution families, there are still no systematic results from Bayesian and empirical Bayesian analysis. In this Ph.D. thesis, some systematic Bayesian and empirical Bayesian analysis results are obtained for the one-parameter truncation distribution families which are nonexponential distribution families. It consists of six main chapters. In Chapter 2, the general forms of conjugate prior distributions are obtained for the two different types of truncation parameter distributions and the particular conjugate priors for the truncated exponential, Pareto and power function distributions are presented. In Chapter 3, the explicit relations between the mixing distributions and the mixture distributions are obtained and the identifiability for the mixture of these trunction parameter distributions is established. Based on these obtained relations, some procedures for estimating the mixing distributions are proposed and studied. In Chapter 4, the explicit analytical expressions of posterior moments for the two general truncation parameter likelihood functions with arbitrary priors are given by using the sufficient statistics for these truncation parameters. In particular, the explicit forms for the posterior mean and variance are presented. In Chapter 5, based on the relations between the Bayes estimators under squared error loss and the marginal distributions, the empirical Bayes estimators are proposed and the asymptotic optimalities of the proposed empirical Bayes estimators are investigated. Finally, in Chapter 6, the problems of empirical Bayes estimation for the truncated exponential distributions and the empirical Bayes rule for selecting the best of exponential populations are discussed and the convergence rates of the proposed empirical Bayes estimators and the empirical Bayes selection rule are established. And in Chapter 7, a location parameter family of gamma distribution, which is not a typical truncation parameter distribution, is considered. The empirical Bayes estimator and the empirical Bayes testing rule for the two-action problem are studied and the convergence rates for the proposed empirical Bayes estimator and the empirical Bayes testing rule are established.

Included in

Mathematics Commons

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