RESARCH ON THE THEORY AND APPLICATIONS OF EFFICIENT BAYES ESTIMATORS IN MULTIVARIATE STATISTICAL MODELS

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In this research project, I surveyed various estimation problems in multivariate statistical models from Theoretical and practical points of view, clarified decision-theoretic results such as admissibility and minimaxity and derived Bayes or shrinkage estimators more efficient than usual procedures.Especially, I made an exhaustive survey research paper covering estimation of mean vectors, mean matrices and covariance matrices of multivariate normal distributions and their extensions to non-normal distributions and estimation of ordered parameters and common parameters. This paper also gives new applicable examples of Stein type shrinkage procedures : one of them is to use empirical Bayes estimators in multicollinearity cases in linear regression models, which provide more efficient and stable estimates than the usual least squares method. The others include not only the derivation of a new variable selection procedure based on the shrinkage method, but also the improved estimators of t … More he noncentrality parameter and the multiple correlation coefficient and their applications to modifying the Mallows statistic and the usual adjusted R-square statistic.Some innovative theoretical results were obtained in this research project. For the estimation of a regression coefficients matrix in a multivariate linear regression model, I derived shrinkage estimators having smaller risks than the least squares estimator and showed the robustness of the improvement within the class of elliptically contoured distributions. This problem is interpreted as a prediction issue in a multivariate mixed linear model and it can be reduced to estimation of ratio of ordered covariance matrices of Wishart distributions. Using this idea and the arguments employed in estimating the covariance matrix, I derived several types of shrinkage estimators improving on the empirical Bayes or Efron-Morris estimator. For the estimation of the covariance matrix in the multivariate linear regression model, on the other hand, I succeeded in resolving a difficult problem, that is, I obtained two methods giving superior and minimax estimators which were constructed by using information contained in the estimator of the regression coefficients. Less