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arXiv · 2206.10167

Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results

Elad Romanov·Gil Kur·Boaz Nadler
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Abstract

Tyler's and Maronna's M-estimators, as well as their regularized variants, are popular robust methods to estimate the scatter or covariance matrix of a multivariate distribution. In this work, we study the non-asymptotic behavior of these estimators, for data sampled from a distribution that satisfies one of the following properties: 1) independent sub-Gaussian entries, up to a linear transformation; 2) log-concave distributions; 3) distributions satisfying a convex concentration property. Our main contribution is the derivation of tight non-asymptotic concentration bounds of these M-estimators around a suitably scaled version of the data sample covariance matrix. Prior to our work, non-asymptotic bounds were derived only for Elliptical and Gaussian distributions. Our proof uses a variety of tools from non asymptotic random matrix theory and high dimensional geometry. Finally, we illustrate the utility of our results on two examples of practical interest: sparse covariance and sparse precision matrix estimation.

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Elad Romanov, Gil Kur, Boaz Nadler. 2022-06-21. Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results. https://doi.org/10.1016/j.jmva.2023.105184

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