# Threshold selection and trimming in extremes

## TL;DR

This paper introduces a new threshold selection method for the Hill estimator in extreme value analysis, using trimming and rescaling techniques to improve tail index estimation and robustness.

## Contribution

It proposes a simple, effective threshold selection procedure that avoids complex tail characteristic estimation and introduces an alternative tail index estimator emphasizing large observations.

## Key findings

- The method performs well in simulations and real insurance data.
- It provides a more robust tail index estimate for lighter tails.
- The approach simplifies threshold selection in extreme value analysis.

## Abstract

We consider removing lower order statistics from the classical Hill estimator in extreme value statistics, and compensating for it by rescaling the remaining terms. Trajectories of these trimmed statistics as a function of the extent of trimming turn out to be quite flat near the optimal threshold value. For the regularly varying case, the classical threshold selection problem in tail estimation is then revisited, 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 which circumvents the estimation of some of the tail characteristics, a problem which is usually the bottleneck in threshold selection. As a by-product, we derive 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 favourable performance and the potential of the proposed method with simulation studies and real insurance data.