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

TL;DR

This paper develops a new approach using concentration of measure and random matrix theory to analyze robust covariance matrix estimators in high-dimensional settings with large sample size and feature dimension.

Contribution

It introduces a fixed point estimator for robust covariance in high dimensions and proves its existence, uniqueness, and spectral properties using semi-metrics and measure concentration.

Findings

Proves the existence and uniqueness of the robust estimator.

Derives the limiting spectral distribution of the estimator.

Shows the estimator's stability via a contracting fixed point approach.

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 $z_i \in \mathbb R^p$ 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.