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

Journal Article

### Authors:

Romain Couillet; Frédéric Pascal; Jack W. Silverstein

### Source:

Journal of Multivariate Analysis, Volume 139, p.56-78 (2015)

### Abstract:

This article demonstrates that the robust scatter matrix estimator $\hat{C}_N ∈ C_{N×N}$ of a multivariate elliptical population $x_1, . . . , x_n ∈ C^N$ 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 $\hat{C}_N ∈ C^{N×N}$ is defined for all n large with probability one and that, under some light hypotheses, $\hat{C}_N − \hat{S}_N → 0$ almost surely in spectral norm, where SˆN 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.

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