Robust $M$-Estimation of Scatter Matrices via Precision Structure Shrinkage

TL;DR

This paper introduces a robust scatter matrix estimator that shrinks the precision matrix towards the identity to improve performance in high-dimensional settings with outliers.

Contribution

It proposes a novel shrinkage-based $M$-estimator for scatter matrices, addressing limitations of existing estimators in high-dimensional, outlier-prone data.

Findings

Estimator improves robustness against clustered outliers

Provides bounds for finite sample breakdown point

Numerical experiments confirm enhanced robustness

Abstract

Maronna's and Tyler's $M$-estimators are among the most widely used robust estimators for scatter matrices. However, when the dimension of observations is relatively high, their performance can substantially deteriorate in certain situations, particularly in the presence of clustered outliers. To address this issue, we propose an estimator that shrinks the estimated precision matrix toward the identity matrix. We derive a sufficient condition for its existence, discuss its statistical interpretation, and establish upper and lower bounds for its additive finite sample breakdown point. Numerical experiments confirm the robustness of the proposed method.