Estimation of a high-dimensional covariance matrix with the Stein loss

TL;DR

This paper develops a decision-theoretic framework for estimating high-dimensional covariance matrices, including singular cases, and introduces unified classes of estimators with dominance results under Stein loss.

Contribution

It unifies the estimation of high-dimensional covariance matrices for both nonsingular and singular cases using a decision-theoretic approach and dominance results.

Findings

Unified classes of estimators with dominance properties

Improved empirical Bayes estimator for high-dimensional covariance matrices

Applicable to both singular and nonsingular cases

Abstract

The problem of estimating a normal covariance matrix is considered from a decision-theoretic point of view, where the dimension of the covariance matrix is larger than the sample size. This paper addresses not only the nonsingular case but also the singular case in terms of the covariance matrix. Based on James and Stein's minimax estimator and on an orthogonally invariant estimator, some classes of estimators are unifiedly defined for any possible ordering on the dimension, the sample size and the rank of the covariance matrix. Unified dominance results on such classes are provided under a Stein-type entropy loss. The unified dominance results are applied to improving on an empirical Bayes estimator of a high-dimensional covariance matrix.