Study of the theory of statistical multivariate analysis and its application to economic analysis

本课题致力于多元正态分布框架下多元分析的理论研究和多元正态假设的研究。在本研究期间，作者出版了《多元统计推断导论》(Kyoritsu Shupban)和《现代数理统计》(Soubunha)两本教材。前者简明扼要地介绍了作者对多元分析的理论框架和最新重要进展的看法。后者是一本关于数理统计理论的可读性教科书，包括研究生水平的材料。关于多元分析的理论研究，作者发表了一篇关于带状多项式的综述论文，继续进行了这方面的研究。在与Yo Sheena合著的一篇论文中，作者发表了一篇关于多元正态总体协方差阵估计中正交不变估计的可容许性的论文。关于协方差矩阵的相等性检验，给出了似然比检验的详细分解。为了研究多元正态分布，作者在正交不变性的框架下讨论了联合高阶矩，并导出了最大不变量。在三阶矩不变的基础上，提出了一种多元正态检验方法。因此，本项目在多变量分析方面取得了有益的成果。另一方面，关于多变量时间序列模型及其在经济分析中的应用的具体结果在本项目中尚未按计划取得。

This project focused on theoretical study of multivariate analysis in the framework of multivariate normal distribution and the investigation of the assumption of multivariate normality. During the period of this study this author published two textbooks, "Introduction to Multivariate Statistical Inference" (Kyoritsu shuppan) and "Modern Mathematical Statistics" (Soubunsha). The former presents this authors's view of the theoretical framework and the recent important developments of multivariate analysis in a concise and self-contained manner. The latter is a readable textbook on the theory of mathematical statistics including graduate level materials. Concerning the theoretical study of multivariate analysis this author published a survey paper on zonal polynomials continuing his study on the subject. In a joint paper with Yo Sheena this author published a paper concerning the admissibility of orthogonally invariant estimator in the estimation of covariance matrix of a multivariate normal population. On the test of equality of covariance matrices this author gave a detailed decomposition of the likelihood ratio test (to appear). For the investigation of the multivariate normality this author discussed joint higher order moments in the framework of orthogonal invariance and derived the maximal invariants. Based on the invariant third order moments this author proposed a test of multivariate normality (to appear). Thus useful results on multivariate analysis have been obtained in this project. On the other hand concrete results on multivariate time series models and its application to economic analysis have not yet been obtained as planned in this project.