Electronic Journal of Statistics

A general decision theory for Huber’s $\epsilon$-contamination model

Mengjie Chen, Chao Gao, and Zhao Ren

Full-text: Open access

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.

Article information

Source
Electron. J. Statist., Volume 10, Number 2 (2016), 3752-3774.

Dates
Received: March 2016
First available in Project Euclid: 6 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1480993453

Digital Object Identifier
doi:10.1214/16-EJS1216

Mathematical Reviews number (MathSciNet)
MR3579675

Zentralblatt MATH identifier
1357.62038

Subjects
Primary: 62C20: Minimax procedures 62G35: Robustness
Secondary: 62H12: Estimation

Keywords
Robust statistics robust testing minimax rate density estimation sparse linear regression trace regression

Citation

Chen, Mengjie; Gao, Chao; Ren, Zhao. A general decision theory for Huber’s $\epsilon$-contamination model. Electron. J. Statist. 10 (2016), no. 2, 3752--3774. doi:10.1214/16-EJS1216. https://projecteuclid.org/euclid.ejs/1480993453


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