Ridge-type Linear Shrinkage Estimation of the Matrix Mean of High-dimensional Normal Distribution

TL;DR

This paper introduces ridge-type linear shrinkage estimators for high-dimensional normal mean matrix estimation, demonstrating their minimax properties and convergence to optimal Bayesian weights, with numerical comparisons showing improved performance.

Contribution

It proposes a novel ridge-type linear shrinkage estimator using ridge estimators for the precision matrix, with estimated weights that are proven to be minimax and converge to Bayesian optimal weights.

Findings

The estimators are minimax under quadratic loss.

Estimated weights converge to Bayesian optimal weights in high dimensions.

Numerical results show superior performance over existing estimators.

Abstract

The estimation of the mean matrix of the multivariate normal distribution is addressed in the high dimensional setting. Efron-Morris-type linear shrinkage estimators based on ridge estimators for the precision matrix instead of the Moore-Penrose generalized inverse are considered, and the weights in the ridge-type linear shrinkage estimators are estimated in terms of minimizing the Stein unbiased risk estimators under the quadratic loss. It is shown that the ridge-type linear shrinkage estimators with the estimated weights are minimax, and that the estimated weights converge to the optimal weights in the Bayesian model with high dimension by using the random matrix theory. The performance of the ridge-type linear shrinkage estimators is numerically compared with the existing estimators including the Efron-Morris and James-Stein estimators.