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

在本研究项目中，我从理论和实践的角度综述了多元统计模型中的各种估计问题，阐明了决策理论的结果，如可容许性和极大极小性，并导出了比通常方法更有效的Bayes或收缩估计，特别是，我做了一篇详尽的综述研究论文，涉及均值向量的估计，多元正态分布的均值阵和协方差阵及其对非正态分布的推广，有序参数和公共参数的估计。本文还给出了Stein型收缩方法的新的应用实例：其中之一是在线性回归模型的多重共线性情况下使用经验Bayes估计，它提供了比通常的最小二乘法更有效和更稳定的估计。另外，还提出了一种新的基于收缩法的变量选择方法，并给出了t的改进估计 ...更多信息 非中心性参数和多重相关系数及其在修正Mallows统计量和通常的调整R方统计量中的应用，取得了一些创新性的理论结果。对于多元线性回归模型中回归系数矩阵的估计，我推导出了比最小二乘估计具有更小风险的收缩估计，并证明了椭圆轮廓分布类内改进的鲁棒性。本文将此问题归结为多元混合线性模型的预测问题，并将其归结为Wishart分布的有序协方差矩阵之比的估计问题。利用这一思想和估计协方差矩阵的参数，我得到了几种类型的收缩估计改进的经验贝叶斯或Efron-Morris估计。另一方面，对于多元线性回归模型中协方差阵的估计，我成功地解决了一个难题，即利用回归系数的估计量中包含的信息，得到了给出上级估计量和Minimax估计量的两种方法。少

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