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

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 breakdown point. Numerical experiments confirm robustness of the proposed method.