We consider estimation of a normal mean matrix under the Frobenius loss. Motivated by the Efron--Morris estimator, a generalization of Stein's prior has been recently developed, which is superharmonic and shrinks the singular values towards zero. The generalized Bayes estimator with respect to this prior is minimax and dominates the maximum likelihood estimator. However, here we show that it is inadmissible by using Brown's condition. Then, we develop two types of priors that provide improved generalized Bayes estimators and examine their performance numerically. The proposed priors attain risk reduction by adding scalar shrinkage or column-wise shrinkage to singular value shrinkage. Parallel results for Bayesian predictive densities are also given.

翻译：我们考虑在Frobenius损失下估计正态均值矩阵的问题。受Efron-Morris估计量的启发，近期发展出一种Stein先验的推广形式，该先验具有超调和性并对奇异值进行零值收缩。基于该先验的广义贝叶斯估计量具有极小极大性，且优于最大似然估计量。然而，本文通过Brown条件证明该估计量是不可容许的。接着，我们构造了两类可提供改进广义贝叶斯估计量的先验，并通过数值实验评估其性能。所提出的先验通过在奇异值收缩基础上加入标量收缩或列向收缩来降低风险。本文还给出了贝叶斯预测密度的并行结果。