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

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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.