Double Shrinkage Priors for a Normal Mean Matrix

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.