Matrix superharmonic priors for Bayes estimation under matrix quadratic loss
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We investigate Bayes estimation of a normal mean matrix under the matrix quadratic loss, which is viewed as a class of loss functions including the Frobenius loss and quadratic loss for each column. First, we derive an unbiased estimate of risk and show that the Efron–Morris estimator is minimax. Next, we introduce a notion of matrix superharmonicity for matrix-variate functions and show that it has 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 include the previously proposed generalization of Stein's prior. Numerical results demonstrate that matrix superharmonic priors work well for low rank matrices.
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