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

在这个研究项目中，我解决了在多元或多维模型中估计参数的几个问题，并从理论和实践的角度得到了新的估计方法，改进了通常的估计方法。本项目的主要成果如下。(1)在线性回归模型中，当解释变量高度相关时，回归系数的最小二乘估计存在估计不稳定、估计误差较大的缺点。在这个多重共线性问题中，我获得了经验贝叶斯岭回归估计，这样它们不仅稳定，而且在最小化均方误差方面改进了最小二乘估计。通过对实际数据的分析，证明了所提方法的实用性。(2)将上述结果推广到多元回归线性模型。(3)在多元线性混合模型的小面积估算问题中，导出了比常规方法更有效的两阶段预测模型，并将其应用于基于神奈川县公示地价数据的小面积估算。(4)对两阶段预测量的均方误差进行了估计，结果表明二阶渐近无偏估计量比不稳定的无偏估计量要好。我在这个项目中处理的其他主题包括以下内容，并为每个主题获得了创新和有用的估计程序：(5) Wishart分布中精度矩阵的估计，(6)限制参数的估计，(7)多变量椭圆轮廓分布中收缩过程改进的鲁棒性，(8)收缩过程均方误差无偏估计的改进，(9)改进协方差矩阵估计的新方法。

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.