In this paper, we introduce a class of improved estimators for the mean parameter matrix of a multivariate normal distribution with an unknown variance-covariance matrix. In particular, the main results of [D.Ch\'etelat and M. T. Wells(2012). Improved Multivariate Normal Mean Estimation with Unknown Covariance when $p$ is Greater than $n$. The Annals of Statistics, Vol. 40, No.6, 3137--3160] are established in their full generalities and we provide the corrected version of their Theorem 2. Specifically, we generalize the existing results in three ways. First, we consider a parameter matrix estimation problem which enclosed as a special case the one about the vector parameter. Second, we propose a class of James-Stein matrix estimators and, we establish a necessary and a sufficient condition for any member of the proposed class to have a finite risk function. Third, we present the conditions for the proposed class of estimators to dominate the maximum likelihood estimator. On the top of these interesting contributions, the additional novelty consists in the fact that, we extend the methods suitable for the vector parameter case and the derived results hold in the classical case as well as in the context of high and ultra-high dimensional data.

翻译：本文针对多元正态分布中均值参数矩阵在未知协方差矩阵情况下的估计问题，提出了一类改进型估计量。具体而言，我们在完全一般性框架下建立了[D. Chételat与M. T. Wells(2012)发表于《统计年鉴》第40卷第6期第3137-3160页的《p大于n时未知协方差矩阵的改进型多元正态均值估计》]的主要结论，并对其定理2给出了校正版本。本研究通过三个维度拓展了现有成果：首先，我们将向量参数估计问题作为特例纳入参数矩阵估计框架；其次，提出了一类James-Stein型矩阵估计量，建立了该类估计量具有有限风险函数的必要与充分条件；最后，给出了所提估计类优于极大似然估计的条件。在上述重要贡献之外，本研究的另一创新在于：将适用于向量参数情形的方法进行推广，所得结论既适用于经典场景，也适用于高维及超高维数据情形。