Stein phenomena and further development on shrinkage methods

众所周知，在假设正态性和适当的损失函数的情况下，样本均值和样本协方差矩阵是不可接受的（这被称为斯坦因效应或收缩法）。自从发现这一现象以来，已经有大量关于这一主题的研究。此外，所谓的Stein's unbiased risk estimate （SURE）也受到了广泛的关注，它最初是为了评估所考虑的估计者的风险函数而提出的一种方法，超出了最初的目的。虽然近年来有了这方面的新进展，但基于收缩法的复杂结构多变量分析仍存在许多问题需要考虑。本研究的目的是发展复杂多元模型的收缩方法。在这一年里，我们研究了以下统计模型并开发了收缩估计器：(1)我们考虑了估计两种增长曲线模型的共同回归系数矩阵的问题，并提出了新的收缩估计器。此外，我们还通过数值实验证明了我们的估计器比已知的估计器有更好的性能。(2)考虑了在平方损失函数下多元正态分布精度矩阵的估计问题，得到了改进的估计量。(3)应用对称锥理论和广义Wishart分布，提出了改进的估计方法，包括复数Wishart分布协方差矩阵的估计问题。

It is well-known that the sample mean and the sample covariance matrix are inadmissible under assumption of normality and appropriate loss function (This is called Stein effect or shrinkage methods.). Since discovery of this phenomenon, there has been a large body of studies on this topic. Furthermore, there has been a broad attention on so called Stein's unbiased risk estimate (SURE), which has been originally proposed for a method to evaluate risk function of estimators under consideration, beyond the original purpose. Although such new development has been reported recently, there are many problems in multivariate analysis with complex structure to be considered based on shrinkage method. The purpose of our research is to develop shrinkage method in complex multivariate model.In this year we investigate on following statistical models and develop shrinkage estimators :(1)We have considered the problem of estimating common regression coefficient matrix of two growth curve models and proposed new shrinkage estimators. Furthermore, we demonstrate numerical experiment to indicate that our proposed estimators have better performance over known estimators.(2)We have considered the problem of estimating a precision matrix of the multivariate normal distribution under the squared loss function and obtained improved estimators.(3)Applying theory of symmetric cones and generalized Wishart distributions, we have developed improved estimation method including the problem of estimating covariance matrix of complex Wishart distribution.