A Concentration of Measure and Random Matrix Approach to Large Dimensional Robust Statistics

This article studies the robust covariance matrix estimation of a data collection X = (x1,ldots,xn) with xi = √ τi zi + m, where zi ∈ mathbb Rp is a textitconcentrated vector (e.g., an elliptical random vector), m∈ mathbb Rp a deterministic signal and τi∈ mathbb R a scalar perturbation of possibly large amplitude, under the assumption where both n and p are large. This estimator is defined as the fixed point of a function which we show is contracting for a so-called textitstable semi-metric. We exploit this semi-metric along with concentration of measure arguments to prove the existence and uniqueness of the robust estimator as well as evaluate its limiting spectral distribution.