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

Abstract

This article studies the \emph{robust covariance matrix estimation} of a data collection $X=(x_1,\dots ,x_n)$ with $x_i={\sqrt{\tau }}_i{z}_i+m$, where $z_i\in {\mathbb{R}}^p$ is a \textit{concentrated vector} (e.g., an elliptical random vector), $m\in {\mathbb{R}}^p$ a deterministic signal and ${\tau }_i\in \mathbb{R}$ 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 \textit{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.

28 pages, 1 figure

Cited by in corpus (1)

• Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results