A Concentration of Measure and Random Matrix Approach to Large Dimensional Robust Statistics
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This article studies the robust covariance matrix estimation of a data collection X = (x_1,...,x_n) with x_i = √(τ)_i z_i + m, where z_i ∈R^p is a concentrated vector (e.g., an elliptical random vector), m∈R^p a deterministic signal and τ_i∈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 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.
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