A Concentration of Measure and Random Matrix Approach to Large Dimensional Robust Statistics

• URL: http://arxiv.org/abs/2006.09728v2
• Date: Sat, 9 Apr 2022 03:10:21 GMT
• Title: A Concentration of Measure and Random Matrix Approach to Large Dimensional Robust Statistics
• Authors: Cosme Louart and Romain Couillet
• Abstract summary: This article studies the emphrobust covariance matrix estimation of a data collection $X = (x_1,ldots,x_n)$ with $x_i = sqrt tau_i z_i + m$. 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.
• Score: 45.24358490877106
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This article studies the \emph{robust covariance matrix estimation} of a data collection $X = (x_1,\ldots,x_n)$ with $x_i = \sqrt \tau_i z_i + m$, where $z_i \in \mathbb R^p$ is a \textit{concentrated vector} (e.g., an elliptical random vector), $m\in \mathbb R^p$ a deterministic signal and $\tau_i\in \mathbb R$ 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 \textit{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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