Matrix norm shrinkage estimators and priors

We develop a class of minimax estimators for a normal mean matrix under the Frobenius loss, which generalizes the James–Stein and Efron–Morris estimators. It shrinks the Schatten norm towards zero and works well for low-rank matrices. We also propose a class of superharmonic priors based on the Schatten norm, which generalizes Stein’s prior and the singular value shrinkage prior. The generalized Bayes estimators and Bayesian predictive densities with respect to these priors are minimax. We examine the performance of the proposed estimators and priors in simulation.