Robust Modifications of U-statistics and Applications to Covariance Estimation Problems

Abstract

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 tight 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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Cited by in corpus (4)

• Robust covariance estimation under $L_4-L_2$ norm equivalence
• User-Friendly Covariance Estimation for Heavy-Tailed Distributions
• Robust Estimation of Covariance Matrices: Adversarial Contamination and Beyond
• Moment inequalities for matrix-valued U-statistics of order 2