Robust covariance estimation and linear shrinkage in the large dimensional regime

The article studies two regularized robust estimators of scatter matrices proposed in parallel in [1] and [2], based on Tyler´s robust M-estimator [3] and on Ledoit and Wolf´s shrinkage covariance matrix estimator [4]. These hybrid estimators convey robustness to outliers or impulsive samples and small sample size adequacy to the classical sample covariance matrix estimator. We consider here the case of i.i.d. elliptical zero mean samples in the regime where both sample and population sizes are large. We prove that the above estimators behave similar to well-understood random matrix models, which allows us to derive optimal shrinkage strategies to estimate the population scatter matrix, largely improving existing methods.