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

Abstract : This article demonstrates that the robust scatter matrix estimator 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 C N ∈ C N ×N is defined for all n large with probability one and that, under some light hypotheses, ˆ C N − ˆ S N → 0 almost surely in spectral norm, wherê S N follows a classical random matrix model. As a corollary, the limiting eigenvalue distribution of C N is derived. This analysis finds applications in the fields of statistical inference and signal processing.
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Romain Couillet, Frédéric Pascal, Jack W. Silverstein. The random matrix regime of Maronna’s M-estimator with elliptically distributed samples. Journal of Multivariate Analysis, Elsevier, 2015, 139, pp.56-78. ⟨10.1016/j.jmva.2015.02.020⟩. ⟨hal-01242488⟩

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