Topics in objective bayesian methodology and spatio-temporal models

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] Three distinct but related topics contribute my work in objective Bayesian methodology and spatio-temporal models. This dissertation starts with the study of a class of objective priors on normal means and variance in a multivariate normal model. The availability of the exact matching priors, such as the right Haar priors, for many parameters is substantiated and the inferential properties are explored. The remaining parts focus on the special multivariate normal models which are Gaussian Markov Random Fields (GMRFs). An intrinsic auto-regressive process (IAR), interpreted as a limiting type of GMRFs appears apealing in estimating smoothing functions. We propose the nonparametric Bayesian hierarchical IAR methods to smooth the discrete hazard rates. Adaptive GMRFs are also used to capture local smoothness. In another perspective, GMRFs are also popular in the field of spatial statistics. One of such well known GMRFs is the conditional auto-regression models (CAR). Motivated by the importance of small-area variation for the development and implementation of medical, educational and economic interventions, we develop a series of Bayesian hierarchical survival models to study the breast cancer incidences in Iowa and consider the spatially correlated frailties using CARs. Due to the limited monitoring time and improvement of medical research, cure rate models are ripe in breast cancer studies. We then propose several semi-parametric Bayesian cure rate models accounting for cure fractions. To release the boundary assumptions in CARs, multi-level spatial effects are modeled via the thin-plate spline (TPS), which also belongs to the GMRFs. Data analyzed were recorded by Surveillance, Epidemiology, and End Results (SEER) registries. The monitoring time window was from year 1991 to 1999.