We investigate estimation of a normal mean matrix under the matrix quadratic loss. Improved estimation under the matrix quadratic loss implies improved estimation of any linear combination of the columns. First, an unbiased estimate of risk is derived and the Efron--Morris estimator is shown to be minimax. Next, a notion of \textit{matrix superharmonicity} for matrix-variate functions is introduced and shown to have analogous properties with usual superharmonic functions, which may be of independent interest. Then, we show that the generalized Bayes estimator with respect to a matrix superharmonic prior is minimax. We also provide a class of matrix superharmonic priors that includes the previously proposed generalization of Stein's prior. Numerical results demonstrate that matrix superharmonic priors work well for low rank matrices.

翻译：我们调查矩阵二次损失下正常平均矩阵的估计。 改进矩阵二次损失的估计意味着改进对各栏线性组合的估计。 首先,得出了无偏向的风险估计,而Efron-Morris估计值显示为微量。 其次,引入了矩阵- 变异功能的\ textit{matrix超级协调性}概念,并显示其具有类似特性,具有通常超和谐功能,可能具有独立的兴趣。 然后,我们表明,通用的贝亚斯测算器在之前的基团超协调前是小型的。 我们还提供了一组矩阵超和谐前科,其中包括先前提议的斯坦先前的通用。 数值结果显示,矩阵超和谐前科对低级矩阵效果良好。