  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 conditions and discuss its
minimax optimal rate in the Hall class. We introduce the so-called trimmed Hill
plot, which can be used to select the number of top order statistics to trim.
We also develop an automatic, data-driven procedure for the choice of trimming.
This results in a new type of robust estimator that can {\em adapt} to the
unknown level of contamination in the extremes. As a by-product we also obtain
a methodology for identifying extreme outliers in heavy tailed data. The
competitive performance of the trimmed Hill and adaptive trimmed Hill
estimators is illustrated with simulations.
