Large-dimensional behavior of regularized Maronna's M-estimators of covariance matrices

TL;DR

This paper analyzes the large-dimensional behavior of regularized Maronna's M-estimators of covariance matrices, providing theoretical insights into their robustness and optimal regularization in high-dimensional settings.

Contribution

It offers a random matrix theory-based characterization of these estimators' asymptotic performance and introduces a data-driven method for optimal regularization parameter selection.

Findings

Many regularized-Maronna estimators share similar asymptotic performance without outliers.

A data-driven method for selecting the optimal regularization parameter is proposed.

Differences in robustness to outliers are characterized for regularized and non-regularized estimators.

Abstract

Robust estimators of large covariance matrices are considered, comprising regularized (linear shrinkage) modifications of Maronna's classical M-estimators. These estimators provide robustness to outliers, while simultaneously being well-defined when the number of samples does not exceed the number of variables. By applying tools from random matrix theory, we characterize the asymptotic performance of such estimators when the numbers of samples and variables grow large together. In particular, our results show that, when outliers are absent, many estimators of the regularized-Maronna type share the same asymptotic performance, and for these estimators we present a data-driven method for choosing the asymptotically optimal regularization parameter with respect to a quadratic loss. Robustness in the presence of outliers is then studied: in the non-regularized case, a large-dimensional robustness metric is proposed, and explicitly computed for two particular types of estimators, exhibiting interesting differences depending on the underlying contamination model. The impact of outliers in regularized estimators is then studied, with interesting differences with respect to the non-regularized case, leading to new practical insights on the choice of particular estimators.