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 $X = (x_1,\ldots,x_n)$ with $x_i = \sqrt \tau_i z_i + m$, where $z_i \in \mathbb R<sup>p$ is a \textit{concentrated vector} (e.g., an elliptical random vector), $m\in \mathbb R<sup>p$ 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.