Hostname: page-component-cb9f654ff-mx8w7 Total loading time: 0 Render date: 2025-08-22T13:45:06.044Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of theprobability-weighted moments and penultimate approximation

Published online by Cambridge University Press:  15 May 2003

Jean Diebolt ,
Armelle Guillou  and
Rym Worms
Jean Diebolt
Affiliation:
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.
Armelle Guillou
Affiliation:
Université Paris VI, Laboratoire de Statistique Théorique et Appliquée, Boîte 158, 175 rue du Chevaleret, 75013 Paris, France; guillou@ccr.jussieu.fr.
Rym Worms
Affiliation:
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.
Get access

Abstract

The P.O.T. (Peaks-Over-Threshold) approachconsists of using the Generalized ParetoDistribution (GPD)to approximate the distribution of excesses over a threshold.We use the probability-weighted momentsto estimate the parameters of the approximating distribution.We study the asymptotic behaviour ofthese estimators (in particular their asymptotic bias) and also thefunctional bias of the GPD as an estimate of thedistribution function of the excesses. We adapt penultimateapproximation results to the case where parameters are estimated.

Keywords

Extreme valuesdomain of attractionexcessesGeneralized Pareto Distributionprobability-weighted momentspenultimate approximation.

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 219 - 238
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Balkema, A. and de Haan, L., Residual life time at a great age. Ann. Probab. 2 (1974) 792-801. CrossRef
Beirlant, J., Dierckx, G., Goegebeur, Y. and Matthys, G., Tail index estimation and an exponential regression model. Extremes 2 (1999) 177-200. CrossRef
Cohen, J.P., Convergence rates for the ultimate and penultimate approximations in extreme-value theory. Adv. Appl. Prob. 14 (1982) 833-854. CrossRef
Dekkers, A.L.M. and de Haan, L., On the estimation of the extreme-value index and large quantile estimation. Ann. Statist. 17 (1989) 1795-1832. CrossRef
Diebolt, J., Durbec, V., El Aroui, M.A. and Villain, B., Estimation of extreme quantiles: Empirical tools for methods assessment and comparison. Int. J. Reliability Quality Safety Engrg. 7 (2000) 75-94. CrossRef
J. Diebolt and M.A. El Aroui, On the use of Peaks over Threshold methods for estimating out-of-sample quantiles. Comput. Statist. Data Anal. (to appear).
Drees, H., A general class of estimators of the extreme value index. J. Statist. Plann. Inf. 66 (1998) 95-112. CrossRef
Einmahl, U. and Mason, D.M., Approximation to permutation and exchangeable processes. J. Theor. Probab. 5 (1992) 101-126. CrossRef
Feuerverger, A. and Hall, P., Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Statist. 27 (1999) 760-781.
J. Galambos, Asymptotic theory of extreme order statistics. Krieger, Malabar, Florida (1978).
Gnedenko, B.V., Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44 (1943) 423-453. CrossRef
Gomes, M.I., Penultimate limiting forms in extreme value theory. Ann. Inst. Stat. Math. 36 (1984) 71-85. CrossRef
Gomes, I. and de Haan, L., Approximation by penultimate extreme value distributions. Extremes 2 (2000) 71-85. CrossRef
M.I. Gomes and D.D. Pestana, Non standard domains of attraction and rates of convergence. John Wiley & Sons (1987) 467-477.
de Haan, L. and Rootzén, H., On the estimation of high quantiles. J. Statist. Plann. Infer. 35 (1993) 1-13. CrossRef
Hosking, J. and Wallis, J., Parameter and quantile estimation for the Generalized Pareto Distribution. Technometrics 29 (1987) 339-349. CrossRef
Pickands III, J., Statistical inference using extreme order statistics. Ann. Statist. 3 (1975) 119-131.
G.R. Shorack and J.A. Wellner, Empirical Processes with Applications to Statistics. Wiley, New York (1986).
Smith, R.L., Estimating tails of probability distributions. Ann. Statist. 15 (1987) 1174-1207. CrossRef
R. Worms, Vitesses de convergence pour l'approximation des queues de distributions. Thèse de doctorat de l'Université de Marne-la-Vallée (2000).
Worms, R., Penultimate approximation for the distribution of the excesses. ESAIM: P&S 6 (2002) 21-31. CrossRef