A concentration of measure and random matrix approach to large-dimensional robust statistics

Abstract

This article studies the robust covariance matrix estimation of a data collection X=(x1,…,xn) with xi=τ izi+m, where zi∈Rp is a concentrated vector (e.g., an elliptical random vector), m∈Rp a deterministic signal and τi∈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 stable 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.