Two Bayesian Methods Based On The Minimization Of The Risk Functions

This paper consists of four chapters.The introduction is located in Chapter 1. In section 1, we introduce the history, development and virtues of the empirical Bayes. In section 2, we introduce theorems and lemmas that are used in Chapters 2 and 3.In Chapter 2, we consider a stepwise con?dence interval procedure under unknown variances based on asymmetric loss for toxicological evaluation. One of the most important issues in toxicity studies is the identi?cation of the equivalence of treatments with a placebo. In the toxicity studies, it is unreasonable to adopt classic symmetric con?dence interval. For the same con?dence level, asymmetric con?dence interval can severely control the toxicity and also properly control the family-wise error rate(FWER). Recently Tao, Tang & Shi(2010) developed a new procedure based on an asymmetric loss function. However, their procedure is somewhat unsatisfactory because it assumes that the variances of various dose levels are known. This assumption is restrictive for some applications. There are three steps for the construction of the asymmetric minimax con?dence interval. We start with the construction of the minimax lower con?dence bound by minimizing the posterior risk. The minimax upper con?dence bound is gained by the same way.Furthermore, the stepwise asymmetric minimax con?dence interval is constructed.In this chapter, we propose an improved approach based on asymmetric con?dence intervals without the restrictive assumption of known variances. The asymmetry guarantees reliability in the sense that the FWER is well controlled. Furthermore,simulation studies showed that the proposed procedure is quite robust in the sense that it can control the FWER even when the homogeneity and/or the normality assumptions are not satis?ed.In Chapter 3, we consider the shrinkage estimator based on the minimization of the risk. Shrinkage estimators in the hierarchical models have profound impacts in statistics and in scienti?c and engineering applications. The seminal work by James and Stein(1961) and Stein(1962) makes shrinkage estimation become one major focus for hierarchical models. There has been substantial research toward the risk of shrinkage estimators for the homoscedastic hierarchical normal models. For the heteroscedastic normal model, Xie et al.(2012) proposed a class of shrinkage estimators based on the Stein.s unbiased estimate of risk(SURE). Their method is one step method to minimize the unbiased estimate of the risk. The shrinkage estimators we propose is based on the iteration to minimize the posterior estimate of the risk. When the iteration stops, the posterior risk estimate is called the posterior mean minimum square risk estimate(denoted by PMMSRE). We study asymptotic properties of the proposed shrinkage estimators based on the PMMSRE method as the number of means to be estimated grows to in?nity. We emphasize that though the form of our PMMSRE estimators is partially obtained through a normal model at the sampling level, their optimality properties do not heavily depend on such distributional assumptions. We apply the methods to real data and obtain satisfactory results.The related proofs of the theorems in the Chapters 2 and 3 are provided in Chapter 4.