How Can Non-invariant Statistics Work in Our Benefit in the Semi-parametric Estimation of Parameters of Rare Events

Abstract

In this article, and in a context of regularly varying tails, we suggest a simple generalization of the classical Hill estimator of a positive tail index γ. Such an estimator is merely the Hill estimator associated to artificially shifted data. The shift a imposed to the data is the tuning parameter of the statistical procedure of estimation. Such a tuning parameter enables us, in a great diversity of situations, to reduce the main component of the bias of Hill's estimator, getting thus estimates with stable sample paths around the target value γ, and with small mean squared error.

Keywords:

• Statistical theory of extremes
• Semi-parametric estimation
• Non-invariant tall index estimators
• Ams 1991 Subject Classifications: Primary: 62G05
• 62E25
• Secondary: 62F35

Acknowledgment

This research was partially supported by FCT/POCTI/FEDER.