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August 2019 User-Friendly Covariance Estimation for Heavy-Tailed Distributions
Yuan Ke, Stanislav Minsker, Zhao Ren, Qiang Sun, Wen-Xin Zhou
Statist. Sci. 34(3): 454-471 (August 2019). DOI: 10.1214/19-STS711

Abstract

We provide a survey of recent results on covariance estimation for heavy-tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we introduce elementwise and spectrumwise truncation operators, as well as their $M$-estimator counterparts, to robustify the sample covariance matrix. Different from the classical notion of robustness that is characterized by the breakdown property, we focus on the tail robustness which is evidenced by the connection between nonasymptotic deviation and confidence level. The key insight is that estimators should adapt to the sample size, dimensionality and noise level to achieve optimal tradeoff between bias and robustness. Furthermore, to facilitate practical implementation, we propose data-driven procedures that automatically calibrate the tuning parameters. We demonstrate their applications to a series of structured models in high dimensions, including the bandable and low-rank covariance matrices and sparse precision matrices. Numerical studies lend strong support to the proposed methods.

Citation

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Yuan Ke. Stanislav Minsker. Zhao Ren. Qiang Sun. Wen-Xin Zhou. "User-Friendly Covariance Estimation for Heavy-Tailed Distributions." Statist. Sci. 34 (3) 454 - 471, August 2019. https://doi.org/10.1214/19-STS711

Information

Published: August 2019
First available in Project Euclid: 11 October 2019

zbMATH: 07162132
MathSciNet: MR4017523
Digital Object Identifier: 10.1214/19-STS711

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.34 • No. 3 • August 2019
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