Shrinkage estimation with a matrix loss function

TL;DR

This paper introduces a matrix-based shrinkage estimator for multivariate normal means, demonstrating its superiority over the maximum likelihood estimator under certain conditions using a matrix quadratic error loss.

Contribution

It proposes a matrix James-Stein estimator with a tuning constant, extending shrinkage estimation to matrix-valued parameters and showing its dominance over MLE.

Findings

The matrix James-Stein estimator dominates MLE for some tuning constants when n ≥ 3.

The approach extends to other shrinkage estimators and settings.

The estimator improves estimation accuracy under matrix quadratic loss.

Abstract

Consider estimating the n by p matrix of means of an n by p matrix of independent normally distributed observations with constant variance, where the performance of an estimator is judged using a p by p matrix quadratic error loss function. A matrix version of the James-Stein estimator is proposed, depending on a tuning constant. It is shown to dominate the usual maximum likelihood estimator for some choices of of the tuning constant when n is greater than or equal to 3. This result also extends to other shrinkage estimators and settings.