Research on derivations of Bayes estimators with decision-theoretical optimality and their applications

贝叶斯程序的有用性最近已经从实际方面得到了承认。在这个研究项目中，我已经从决策理论的角度证明了贝叶斯过程在几个统计问题中的最优性以及在应用中的有效性。1)在多元正态分布的均值向量估计中，给出了先验分布的刻画，使得所得到的贝叶斯估计是极小极大估计和/或可容许的。2)在多元线性混合模型协方差矩阵分量的估计中，通过截断方法建立了改进的统一理论。这个问题涉及到在不等式约束下协方差矩阵的估计问题。我考虑了几个参数约束下的估计问题，证明了贝叶斯估计的优势结果。得到了高维情形下协方差矩阵的经验Bayes估计，证明了理论上的最优性以及在数据分析中的实用性。3)在嵌套误差回归模型中，给出了选择解释变量的信息准则。这个模型在小区域问题中是有用的，我构造了小区域均值的渐近校正的可信区间和线性假设的渐近校正的检验统计量。

The usefulness of the Bayesian procedures has been recently recognized from practical aspects. In this research project, I have shown the optimality of the Bayesian procedures from a decision-theoretic view point in several statistical problems as well as the usefulness in applications. The details are given below:1) In the estimation of a mean vector of a multivariate normal distribution, the characterization of the prior distributions has been given so that the resulting Bayes estimator is minimax and/or admissible. When the prior distribution has a hierarchical structure, I have derived conditions under which the hierarchical Bayes estimators are minimax.2) In the estimation of the component of covariance matrix in a multivariate linear mixed model, I have established a unified theory for the improvement through the truncated method. This problem is related to the estimation of the covariance matrices under the inequality restriction. I have considered several estimation problems under parametric restrictions and have shown the dominance results of Bayesian estimators. Also I have obtained the empirical Bayes estimator of the covariance matrix in the high dimensional cases and shown the theoretical optimality as well as the practical usefulness in data analysis.3) In the nested error regression model, I have derived the information criterion for selecting explanatory variables. This model is useful in the small area problem, and I have constructed an asymptotically corrected confidence interval of the small area mean and an asymptotically corrected test statistic for the linear hypothesis.