RESARCH ON NEW DEVELOPMENTS OF ESTIMATION THEORY AND THEIR APPLICATIONS IN MULTI-DIMENSIONAL STATISTICAL MODELS

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In this research project, I addressed several issues of estimating parameters in multivariate or multi-dimensional models and obtained new estimation procedures which improve on usual estimators from theoretical and practical points of view. The main results of this project are given below.(1)When explanatory variables are highly correlated in a linear regression model, the least squares estimators of the regression coefficients have the drawback that their estimates are not stable and their estimation errors are considerably large. In this multicollinearity problem, I obtained empirical Bayes ridge regression estimators such that they are not only stable, but also improving on the least squares estimators in terms of minimizing the mean squared errors. I demonstrate that the proposed methods are practically useful through analyzing real data. (2)The results stated above were extended to multivariate regression linear models. (3)In small area estimation problems in multivariate linear mixed models, two-stage predictors more efficient than usual procedures were derived and applied to the small area estimation based on the posted land price data in Kanagawa prefecture. (4)Estimation of mean squared errors of the two-stage predictors was treated, and it was shown that second-order asymptotic unbiased estimators of the mean squared errors are better than the unbiased estimators being often instable. Other topics I dealt with in this project include the following, and innovative and useful estimation procedures were obtained for each topic: (5)estimation of a precision matrix in the Wishart distribution, (6)estimation of restricted parameters, (7)robustness of improvement of shrinkage procedures in multivariate elliptically contoured distributions, (8)improvement on unbiased estimators of mean squared errors of shrinkage procedures, (9)a new approach to improvement in estimation of a covariance matrix.