Estimation under matrix quadratic loss and matrix superharmonicity

Abstract

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 under the quadratic loss. First, an unbiased estimate of risk is derived and the Efron-Morris estimator is shown to be minimax. Next, a notion of matrix superharmonicity for matrix-variate functions is introduced and shown to have properties analogous to those of the usual superharmonic functions, which may be of independent interest. Then, it is shown 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.