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December 2018 Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries
Stanislav Minsker
Ann. Statist. 46(6A): 2871-2903 (December 2018). DOI: 10.1214/17-AOS1642

Abstract

Estimation of the covariance matrix has attracted a lot of attention of the statistical research community over the years, partially due to important applications such as principal component analysis. However, frequently used empirical covariance estimator, and its modifications, is very sensitive to the presence of outliers in the data. As P. Huber wrote [Ann. Math. Stat. 35 (1964) 73–101], “…This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): what happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance….” Motivated by Tukey’s question, we develop a new estimator of the (element-wise) mean of a random matrix, which includes covariance estimation problem as a special case. Assuming that the entries of a matrix possess only finite second moment, this new estimator admits sub-Gaussian or sub-exponential concentration around the unknown mean in the operator norm. We explain the key ideas behind our construction, and discuss applications to covariance estimation and matrix completion problems.

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Stanislav Minsker. "Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries." Ann. Statist. 46 (6A) 2871 - 2903, December 2018. https://doi.org/10.1214/17-AOS1642

Information

Received: 1 September 2016; Revised: 1 August 2017; Published: December 2018
First available in Project Euclid: 7 September 2018

zbMATH: 06968602
MathSciNet: MR3851758
Digital Object Identifier: 10.1214/17-AOS1642

Subjects:
Primary: 60B20, 62G35
Secondary: 62H12

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6A • December 2018
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