  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.
