  We introduce a trimmed version of the Hill estimator for the index of a
heavy-tailed distribution, which is robust to perturbations in the extreme
order statistics. In the ideal Pareto setting, the estimator is essentially
finite-sample efficient among all unbiased estimators with a given strict upper
break-down point. For general heavy-tailed models, we establish the asymptotic
normality of the estimator under second order regular variation conditions and
also show it is minimax rate-optimal in the Hall class of distributions. We
also develop an automatic, data-driven method for the choice of the trimming
parameter which yields a new type of robust estimator that can adapt to the
unknown level of contamination in the extremes. This adaptive robustness
property makes our estimator particularly appealing and superior to other
robust estimators in the setting where the extremes of the data are
contaminated. As an important application of the data-driven selection of the
trimming parameters, we obtain a methodology for the principled identification
of extreme outliers in heavy tailed data. Indeed, the method has been shown to
correctly identify the number of outliers in the previously explored Condroz
data set.
