Equivariant minimax dominators of the MLE in the array normal model

TL;DR

This paper introduces a new class of estimators for the array normal model that are minimax and outperform the maximum likelihood estimator, with practical improvements demonstrated through numerical comparisons.

Contribution

It establishes the optimality and minimax properties of a new equivariant estimator in the array normal model, extending classical covariance estimation results.

Findings

The UMREE is minimax and dominates the MLE.

Orthogonally equivariant modifications improve estimator performance.

Numerical results show substantially lower risks for the proposed estimators.

Abstract

Inference about dependencies in a multiway data array can be made using the array normal model, which corresponds to the class of multivariate normal distributions with separable covariance matrices. Maximum likelihood and Bayesian methods for inference in the array normal model have appeared in the literature, but there have not been any results concerning the optimality properties of such estimators. In this article, we obtain results for the array normal model that are analogous to some classical results concerning covariance estimation for the multivariate normal model. We show that under a lower triangular product group, a uniformly minimum risk equivariant estimator (UMREE) can be obtained via a generalized Bayes procedure. Although this UMREE is minimax and dominates the MLE, it can be improved upon via an orthogonally equivariant modification. Numerical comparisons of the risks of these estimators show that the equivariant estimators can have substantially lower risks than the MLE.