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

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 Scheff{\'e} 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, Scheff{\'e} estimate, also enjoys an optimal rate in the exponent of the testing error, which may be of independent interest.