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 $\mathit{X}=({\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}})$ with ${\mathit{x}}_{\mathit{i}}={\sqrt{\mathit{\tau }}}_{\mathit{i}}{\mathit{z}}_{\mathit{i}}+\mathit{m}$, where ${\mathit{z}}_{\mathit{i}}\in {\mathbb{R}}^{\mathit{p}}$ is a concentrated vector (e.g., an elliptical random vector), $\mathit{m}\in {\mathbb{R}}^{\mathit{p}}$ a deterministic signal and ${\mathit{\tau }}_{\mathit{i}}\in \mathbb{R}$ 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 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.