The Random Matrix Regime of Maronna's M-estimator with elliptically distributed samples

TL;DR

This paper analyzes Maronna's robust scatter matrix estimator for elliptical distributions, showing it behaves like a classical random matrix model in high-dimensional regimes and deriving its eigenvalue distribution.

Contribution

It establishes the asymptotic equivalence of Maronna's estimator to a well-known random matrix model in high dimensions, with convergence results and eigenvalue distribution.

Findings

$ orm{C_N - S_N} o 0$ almost surely in spectral norm

Eigenvalue distribution of $C_N$ derived in the limit

Estimator behaves like classical random matrix models asymptotically

Abstract

This article demonstrates that the robust scatter matrix estimator $\hat{C}_N\in {\mathbb C}^{N\times N}$ of a multivariate elliptical population $x_1,\ldots,x_n\in {\mathbb C}^N$ originally proposed by Maronna in 1976, and defined as the solution (when existent) of an implicit equation, behaves similar to a well-known random matrix model in the limiting regime where the population $N$ and sample $n$ sizes grow at the same speed. We show precisely that $\hat{C}_N\in{\mathbb C}^{N\times N}$ is defined for all $n$ large with probability one and that, under some light hypotheses, $\Vert \hat{C}_N-\hat{S}_N\Vert\to 0$ almost surely in spectral norm, where $\hat{S}_N$ follows a classical random matrix model. As a corollary, the limiting eigenvalue distribution of $\hat{C}_N$ is derived. This analysis finds applications in the fields of statistical inference and signal processing.