Asymptotic behaviour of the probability-weighted moments and penultimate approximation

¹ Université de Marne-la-Vallée, Équipe d'Analyse et de Mathématiques Appliquées, 5 boulevard Descartes, bâtiment Copernic, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France; diebolt@math.univ-mlv.fr.
² Université Paris VI, Laboratoire de Statistique Théorique et Appliquée, Boîte 158, 175 rue du Chevaleret, 75013 Paris, France; guillou@ccr.jussieu.fr.
³ Université de Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR 6085 du CNRS, Site Colbert, UFR Sciences, 76821 Mont-Saint-Aignan Cedex, France; rym.worms@univ-rouen.fr.

Received: 30 July 2002
Revised: 3 February 2003

Abstract

The P.O.T. (Peaks-Over-Threshold) approach consists of using the Generalized Pareto Distribution (GPD) to approximate the distribution of excesses over a threshold. We use the probability-weighted moments to estimate the parameters of the approximating distribution. We study the asymptotic behaviour of these estimators (in particular their asymptotic bias) and also the functional bias of the GPD as an estimate of the distribution function of the excesses. We adapt penultimate approximation results to the case where parameters are estimated.