Eigenvector distributions and optimal shrinkage estimators for large covariance and precision matrices

TL;DR

This paper investigates Stein's invariant shrinkage estimators for large covariance and precision matrices in high-dimensional settings, establishing their asymptotic behavior and introducing new results on eigenvector distributions.

Contribution

It introduces a novel asymptotic distribution result for non-spiked eigenvectors, enabling analysis of shrinkage estimators in complex high-dimensional models.

Findings

Asymptotic limits of shrinkers are established for various loss functions.

Eigenvector distribution results are derived for non-spiked eigenvectors.

The methods apply to models with nearly arbitrary covariance structures.

Abstract

This paper focuses on investigating Stein's invariant shrinkage estimators for large sample covariance matrices and precision matrices in high-dimensional settings. We consider models that have nearly arbitrary population covariance matrices, including those with potential spikes. By imposing mild technical assumptions, we establish the asymptotic limits of the shrinkers for a wide range of loss functions. A key contribution of this work, enabling the derivation of the limits of the shrinkers, is a novel result concerning the asymptotic distributions of the non-spiked eigenvectors of the sample covariance matrices, which can be of independent interest.