论文标题
矩阵二次损失和矩阵超谐波下的估计
Estimation under matrix quadratic loss and matrix superharmonicity
论文作者
论文摘要
我们研究了基质二次损耗下正常平均基质的估计。矩阵二次损失下的估计改进意味着改进了列的任何线性组合的估计。首先，得出了对风险的无偏估计，并显示出EFRON-MORRIS估计器是最小值。接下来，引入了\ textIt {矩阵超谐波}的概念，用于矩阵变量函数并显示出具有通常超级谐波函数的类似属性，这可能具有独立的兴趣。然后，我们表明，相对于基质超谐波先验的广义贝叶斯估计量是minimax。我们还提供了一类矩阵超级谐波先验，其中包括先前提出的对Stein先验的概括。数值结果表明，矩阵超谐波先验在低级矩阵方面效果很好。
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.
