On the orthogonally equivariant estimators of a covariance matrix

TL;DR

This paper introduces a new orthogonally equivariant estimator for large-dimensional covariance matrices, leveraging the Marčenko-Pastur equation to improve eigenvalue estimation and optimality under Stein loss.

Contribution

The paper develops a novel estimator based on the Marčenko-Pastur equation that is consistent and optimal among orthogonally equivariant estimators for large covariance matrices.

Findings

Estimator is consistent for population eigenvalues

Proposed estimator is optimal under Stein loss

Effective in high-dimensional settings

Abstract

In this note, when the dimension $p$ is large we look into the insight of the Mar$\check{c}$enko-Pastur equation to get an explicit equality relationship, and use the obtained equality to establish a new kind of orthogonally equivariant estimator of the population covariance matrix. Under some regularity conditions, the proposed novel estimators of the population eigenvalues are shown to be consistent for the eigenvalues of population covariance matrix. It is also shown that the proposed estimator is the best orthogonally equivariant estimator of population covariance matrix under the normalized Stein loss function.