Robust Estimates of Covariance Matrices in the Large Dimensional Regime

TL;DR

This paper analyzes the behavior of robust covariance matrix estimators in high-dimensional settings, showing they converge to scaled sample covariance matrices under certain conditions, thus enabling robust statistical methods.

Contribution

It provides a rigorous proof of the spectral norm convergence of robust covariance estimators in large dimensions using random matrix theory.

Findings

Robust estimators converge to scaled sample covariance matrices in spectral norm.

The convergence holds almost surely under certain moment conditions.

Results enable robust adaptations of existing random matrix-based statistical methods.

Abstract

This article studies the limiting behavior of a class of robust population covariance matrix estimators, originally due to Maronna in 1976, in the regime where both the number of available samples and the population size grow large. Using tools from random matrix theory, we prove that, for sample vectors made of independent entries having some moment conditions, the difference between the sample covariance matrix and (a scaled version of) such robust estimator tends to zero in spectral norm, almost surely. This result can be applied to various statistical methods arising from random matrix theory that can be made robust without altering their first order behavior.