Improvements in the Estimation of a Heavy Tail

Abstract

In this paper, and in a context of regularly varying tails, we suggest new tail index estimators, which provide interesting alternatives to the classical Hill estimator of the tail index γ. They incorporate some extra knowledge on the pattern of scaled top order statistics and seem to work generally pretty well in a semi-parametric context, even for cases where a second order condition does not hold or we are outside Hall’s class of models. We shall give particular emphasis to a class of statistics dependent on a tuning parameter τ , which is merely a change in the scale of our data, from X to X/τ . Such a statistic is non-invariant both for changes in location and in scale, but compares favourably with the Hill estimator for a class of models where it is not easy to find competitors to this classic tail index estimator. We thus advance with a slight “controversial” argument: it is always possible to take advantage from a non-invariant estimator, playing with particular tuning parameters — either a change in the location or in the scale of our data —, improving then the overall performance of the classical estimators of extreme events parameters.

Keywords

• statistics of extremes
• semi-parametric estimation
• Monte Carlo methods