Trimming and threshold selection in extremes
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We consider removing lower order statistics from the classical Hill estimator in extreme value statistics, and compensating for it by rescaling the remaining terms. As a function of the extent of trimming, the resulting trajectories are unbiased estimators of the tail index of exact Pareto samples, with lower variance than the Hill estimator when using the same amount of data. For the regularly varying case, the classical threshold selection problem in tail estimation is revisited with this method, both visually via trimmed Hill plots and, for the Hall class, also mathematically via minimizing the expected empirical variance. This leads to a simple threshold selection procedure for the classical Hill estimator, and at the same time also suggests an alternative estimator of the tail index, which assigns more weight to large observations, and works particularly well for relatively lighter tails. A simple ratio statistic routine is suggested to evaluate the goodness of the implied selection of the threshold. We illustrate the performance and potential of the proposed method with simulation studies and real insurance data.
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