Optimclity in multirariate estimation

利用该公司购买的Macintosh II及其数学软件，研究了多元正态总体中均值矩阵、遗传方差、协方差矩阵和隐根的改进估计。得到了广义方差的等效收缩估计器风险的尖锐下界。利用矩阵参数的超几何函数进行数值计算，发现当非零参数矩阵离0不远时，Stein估计量的风险接近于尖锐下界。提出了均值向量或协方差矩阵的序贯收缩估计和两阶段收缩估计，其效果优于普通估计。结果表明，除了二次损失和熵损失外，基于Pitman接近准则得到了对共同均值或广义方差的改进估计量。导出了离群点的局部最优不变量检验方法，并通过仿真证明了该方法的有效性大于普通检验方法的有效性。将双侧指数分布的位置参数估计视为一种非规则分布。证明了在样本中位数附近的一类有序统计量的线性组合中可以找到比极大似然估计量更好的渐近估计量，这是Akahira（1987）的推广。通过仿真比较了在样本量较小的情况下，各估计量的方差值。皮曼估计量的方差最小。然而，我们无法获得渐近方差。本课题所编制的区域多项式计算程序将对其他多变量统计分析问题的求解有一定的指导意义。在此程序的基础上，得到了二元Wishart矩阵的现根分布图。每位研究者积极开展自己的研究课题，并将研究成果向国外报告。少

Using Macintosh II with its software "Mathematics" purchased by this fiend, improved estimators for the mean matrix, genemfted variance, covariance matrix, and the latrnt roots in multivariate normal population were investigated. The sharp lower bound of the risk of the equivalent shrinkage estimator for the generalized variance is obtained. Numerical values are computed, using hypergeometric function of matrix arguments and it was seen that the risk of Stein estimator is close to the sharp lower bound when the matrix of noncennhty parameters is not so far from 0. Sequential shrinkage estimation and two-stage shrinkage estimation for the mean vector or covariance matrix are proposed, which are better than the ordinary estimators. It is shown that improved estimators for the common means or the generalized variance are obtained based on Pitman closeness criterion apart from quadratic loss or entropy loss.The locally best invariant test for outliers is derived and the numerical values of … More the power is shown, by simulation, to be larger than those of the ordinary tests . The estimation of the location parameter of the two-sided exponential distribution is considered as a nonregular distribution. It is shown that asymploticauy better estimator than the maximum likelihood estimator can be found within a class of linear combination of order statistics near the sample median, which is an extension of Akahira(1987). By simulation the values of the variance of each estimator are compared, when the sample size, is small. The Piman estimator has the smallest variance. However we were unable to obtain the asymptotic variance. The computing program of zonal polynomials made in this project will be useful for many other problems in multivariate statiidcal analysis. Graphs of the distribution of the laent roots of the bivariate Wishart matrix is obtained based on this program. Each investigator developed his study in his research theme actively and the results are reported to abroad too. Less