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2016 A general decision theory for Huber’s $\epsilon$-contamination model
Mengjie Chen, Chao Gao, Zhao Ren
Electron. J. Statist. 10(2): 3752-3774 (2016). DOI: 10.1214/16-EJS1216

Abstract

Today’s data pose unprecedented challenges to statisticians. It may be incomplete, corrupted or exposed to some unknown source of contamination. We need new methods and theories to grapple with these challenges. Robust estimation is one of the revived fields with potential to accommodate such complexity and glean useful information from modern datasets. Following our recent work on high dimensional robust covariance matrix estimation, we establish a general decision theory for robust statistics under Huber’s $\epsilon$-contamination model. We propose a solution using Scheffé estimate to a robust two-point testing problem that leads to the construction of robust estimators adaptive to the proportion of contamination. Applying the general theory, we construct robust estimators for nonparametric density estimation, sparse linear regression and low-rank trace regression. We show that these new estimators achieve the minimax rate with optimal dependence on the contamination proportion. This testing procedure, Scheffé estimate, also enjoys an optimal rate in the exponent of the testing error, which may be of independent interest.

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Mengjie Chen. Chao Gao. Zhao Ren. "A general decision theory for Huber’s $\epsilon$-contamination model." Electron. J. Statist. 10 (2) 3752 - 3774, 2016. https://doi.org/10.1214/16-EJS1216

Information

Received: 1 March 2016; Published: 2016
First available in Project Euclid: 6 December 2016

zbMATH: 1357.62038
MathSciNet: MR3579675
Digital Object Identifier: 10.1214/16-EJS1216

Subjects:
Primary: 62C20, 62G35
Secondary: 62H12

Rights: Copyright © 2016 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.10 • No. 2 • 2016
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