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Date of Award

2019

Document Type

Campus Access Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Min Wang

Committee Member 1

Yu Cai

Committee Member 2

Jianping Dong

Committee Member 3

Renfang Jiang

DOI

10.37099/mtu.dc.etdr/928

Abstract

This dissertation consists of five chapters with three distinct but related research projects.

In Chapter 1, we introduce some necessary definitions related to the research work.

In Chapter 2, we develop Bayes factor based testing procedures for a general linear hypothesis of the regression coefficients in the context of the normal linear models. We propose two calibration schemes to deal with asymmetry in information of Bayes factor: (i) by controlling Type I error probability of the Bayes factor and (ii) by balancing Type I and Type II error probabilities of the Bayes factor. We evaluate the finite sample performance of the proposed Bayes factors via simulation studies and a real-data application. Experimental results have shown than the proposed Bayes factors perform well in testing the general linear hypothesis of the regression coefficient.

In Chapter 3, we consider Bayesian quantile analysis for testing constrained hypotheses in linear models, in which the quantiles the parameters satisfy a simple order restriction. We develop a Bayesian hierarchical model based on the specification the asymmetric Laplace distribution for the error component. We propose a non-iterative sampling algorithm in the Expectation-Maximization (EM) structure to generate independently and identically distributed posterior samples from their posterior distributions of the parameters. Then we adopt the Savage-Dickey density ratios to conduct the multiple comparison with simply order constraints. Simulation studies were conducted to compare the finite sample performance of the proposed non-iterative sampling algorithm with the Gibbs sampling algorithm.

In Chapter 4, we consider objective Bayesian analysis for the concordance correlation coefficient (CCC), which is one of the most commonly used metrics to assess agreement of different methods in many practical applications. We develop an objective Bayesian framework for estimating the CCC based a combined use of the multivariate student's t-distribution with noninformative Independence Jeffreys prior for the unknown parameters. Extensive simulation studies are conducted to compare the performance of the proposed Bayesian estimates with the ones under the subjective priors in the literature.

In Chapter 5, we discuss some ongoing projects related to our research work mentioned above and some interesting problems for future work.

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