The random matrix regime of Maronna's M-estimator with elliptically distributed samples

This article demonstrates that the robust scatter matrix estimator (C) over cap (N) is an element of C-NxN of a multivariate elliptical population x(1), ... ,x(n) is an element of 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 (C) over cap (N) is an element of C-NxN is defined for all n large with probability one and that, under some light hypotheses, parallel to(C) over cap (N) - (S) over cap (N)parallel to -> 0 almost surely in spectral norm, where (S) over cap (N) follows a classical random matrix model. As a corollary, the limiting eigenvalue distribution of (C) over cap (N) is derived. This analysis finds applications in the fields of statistical inference and signal processing. (C) 2015 Elsevier Inc. All rights reserved.