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Today's data pose unprecedented challenges to statisticians. It may beincomplete, corrupted or exposed to some unknown source of contamination. Weneed new methods and theories to grapple with these challenges. Robustestimation is one of the revived fields with potential to accommodate suchcomplexity and glean useful information from modern datasets. Following ourrecent work on high dimensional robust covariance matrix estimation, weestablish a general decision theory for robust statistics under Huber's$\epsilon$-contamination model. We propose a solution using Scheff{\'e}estimate to a robust two-point testing problem that leads to the constructionof robust estimators adaptive to the proportion of contamination. Applying thegeneral theory, we construct robust estimators for nonparametric densityestimation, sparse linear regression and low-rank trace regression. We showthat these new estimators achieve the minimax rate with optimal dependence onthe contamination proportion. This testing procedure, Scheff{\'e} estimate,also enjoys an optimal rate in the exponent of the testing error, which may beof independent interest.

A General Decision Theory for Huber's $\epsilon$-Contamination Model