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June 2020 Robust covariance estimation under $L_{4}-L_{2}$ norm equivalence
Shahar Mendelson, Nikita Zhivotovskiy
Ann. Statist. 48(3): 1648-1664 (June 2020). DOI: 10.1214/19-AOS1862

Abstract

Let $X$ be a centered random vector taking values in $\mathbb{R}^{d}$ and let $\Sigma=\mathbb{E}(X\otimes X)$ be its covariance matrix. We show that if $X$ satisfies an $L_{4}-L_{2}$ norm equivalence (sometimes referred to as the bounded kurtosis assumption), there is a covariance estimator $\hat{\Sigma}$ that exhibits almost the same performance one would expect had $X$ been a Gaussian vector. The procedure also improves the current state-of-the-art regarding high probability bounds in the sub-Gaussian case (sharp results were only known in expectation or with constant probability).

In both scenarios the new bounds do not depend explicitly on the dimension $d$, but rather on the effective rank of the covariance matrix $\Sigma$.

Citation

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Shahar Mendelson. Nikita Zhivotovskiy. "Robust covariance estimation under $L_{4}-L_{2}$ norm equivalence." Ann. Statist. 48 (3) 1648 - 1664, June 2020. https://doi.org/10.1214/19-AOS1862

Information

Received: 1 October 2018; Revised: 1 March 2019; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241606
MathSciNet: MR4124338
Digital Object Identifier: 10.1214/19-AOS1862

Subjects:
Primary: 62G35
Secondary: 62G15

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 3 • June 2020
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