🤖 AI Summary
This work addresses the marked degradation in robustness of Maronna’s and Tyler’s M-estimators under high-dimensional observations in the presence of clustered outliers. To mitigate this issue, we propose a novel M-estimation approach that enhances the stability of scatter matrix estimation by shrinking the precision matrix toward the identity matrix. We establish sufficient conditions for the existence of the proposed estimator, derive upper and lower bounds on its breakdown point, and provide both a statistical interpretation and an efficient numerical optimization algorithm for its implementation. Experimental results demonstrate that the proposed method substantially improves robustness and estimation accuracy on high-dimensional data contaminated with clustered outliers.
📝 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.