Summary 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 under the quadratic loss. First, an unbiased estimate of risk is derived and the Efron–Morris estimator is shown to be minimax. Next, a notion of matrix superharmonicity for matrix-variate functions is introduced and shown to have properties analogous to those of the usual superharmonic functions, which may be of independent interest. Then, it is shown 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.

中文翻译：

---

矩阵二次损失和矩阵超谐波下的估计

总结 我们研究了矩阵二次损失下的正态平均矩阵的估计。矩阵二次损失下的改进估计意味着二次损失下列的任何线性组合的改进估计。首先，导出了一个无偏的风险估计，并且 Efron-Morris 估计量显示为极小极大。接下来，介绍了矩阵变函数的矩阵超谐波的概念，并表明它具有类似于通常的超谐波函数的属性，这可能是独立的兴趣。然后，证明了关于矩阵超谐波先验的广义贝叶斯估计是极小极大的。我们还提供了一类矩阵超谐波先验，其中包括先前提出的 Stein 先验推广。