Trimming the Hill estimator: robustness, optimality and adaptivity
Abstract
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.
Cite
@article{arxiv.1705.03088,
title = {Trimming the Hill estimator: robustness, optimality and adaptivity},
author = {Shrijita Bhattacharya and Michael Kallitsis and Stilian Stoev},
journal= {arXiv preprint arXiv:1705.03088},
year = {2017}
}