Open Access
February 2020 Robust modifications of U-statistics and applications to covariance estimation problems
Stanislav Minsker, Xiaohan Wei
Bernoulli 26(1): 694-727 (February 2020). DOI: 10.3150/19-BEJ1149


Let $Y$ be a $d$-dimensional random vector with unknown mean $\mu $ and covariance matrix $\Sigma $. This paper is motivated by the problem of designing an estimator of $\Sigma $ that admits exponential deviation bounds in the operator norm under minimal assumptions on the underlying distribution, such as existence of only 4th moments of the coordinates of $Y$. To address this problem, we propose robust modifications of the operator-valued U-statistics, obtain non-asymptotic guarantees for their performance, and demonstrate the implications of these results to the covariance estimation problem under various structural assumptions.


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Stanislav Minsker. Xiaohan Wei. "Robust modifications of U-statistics and applications to covariance estimation problems." Bernoulli 26 (1) 694 - 727, February 2020.


Received: 1 February 2018; Revised: 1 July 2019; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140514
MathSciNet: MR4036049
Digital Object Identifier: 10.3150/19-BEJ1149

Keywords: Covariance estimation , heavy tails , robust estimators , U-statistics

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 1 • February 2020
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