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

本课题主要对多元正态分布框架下的多元分析进行理论研究，并对多元正态性假设进行研究。在此研究期间，作者出版了《多元统计推断导论》和《现代数理统计》两本教材。前者提出了这个作者的观点的理论框架和多元分析在一个简洁和独立的方式最近的重要发展。后者是一本可读的数理统计理论教科书，包括研究生水平的材料。在多元分析的理论研究方面，作者发表了一篇关于区域多项式的综述论文，继续对这一问题进行研究。在与Yo Sheena合著的论文中，作者讨论了多元正态总体协方差矩阵估计中正交不变估计量的可容许性问题。在协方差矩阵的相等性检验上，作者给出了似然比检验的详细分解（待见）。为了研究多元正态性，本文在正交不变量的框架下讨论了联合高阶矩，并导出了极大不变量。基于不变三阶矩，提出了一种多元正态性的检验方法。因此，本项目在多变量分析方面取得了有益的结果。另一方面，在多元时间序列模型及其在经济分析中的应用方面，本项目尚未按计划取得具体成果。

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.