Proper Bayes and minimax predictive densities related to estimation of a normal mean matrix

This paper deals with the problem of estimating predictive densities of a matrix-variate normal distribution with known covariance matrix. Our main aim is to establish some Bayesian predictive densities related to matricial shrinkage estimators of the normal mean matrix. The Kullback-Leibler loss is used for evaluating decision-theoretic optimality of predictive densities. It is shown that a proper hierarchical prior yields an admissible and minimax predictive density. Also, some minimax predictive densities are derived from superharmonicity of prior densities.