Estimation under matrix quadratic loss and matrix superharmonicity

TL;DR

This paper studies the estimation of a normal mean matrix under matrix quadratic loss, introducing matrix superharmonicity and showing that certain estimators are minimax, with practical benefits for low-rank matrices.

Contribution

It introduces the concept of matrix superharmonicity and demonstrates that generalized Bayes estimators with such priors are minimax, extending Stein's prior to matrix settings.

Findings

The Efron--Morris estimator is minimax under matrix quadratic loss.

Matrix superharmonic priors lead to effective minimax estimators.

Numerical results show advantages for low-rank matrices.

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. First, an unbiased estimate of risk is derived and the Efron--Morris estimator is shown to be minimax. Next, a notion of \textit{matrix superharmonicity} for matrix-variate functions is introduced and shown to have analogous properties with usual superharmonic functions, which may be of independent interest. Then, we show 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.