Hostname: page-component-7479d7b7d-rvbq7 Total loading time: 0 Render date: 2024-07-11T23:25:41.924Z Has data issue: false hasContentIssue false
  • English
  • Français

Rate of convergence for the generalized Pareto approximation of the excesses

Part of: Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

J.-P. Raoult  and
R. Worms
J.-P. Raoult*
Affiliation:
Université de Marne-la-Vallée
R. Worms*
Affiliation:
Université de Marne-la-Vallée
*
∗ Postal address: Université de Marne-la-Vallée, Equipe d'Analyse et de Mathématiques appliquées, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France.
∗ Postal address: Université de Marne-la-Vallée, Equipe d'Analyse et de Mathématiques appliquées, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let F be a distribution function in the domain of attraction of an extreme-value distribution Hγ. If Fu is the distribution function of the excesses over u and Gγ the distribution function of the generalized Pareto distribution, then it is well known that Fu(x) converges to Gγ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅u(x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

Keywords

Generalized Pareto distributionexcessesrate of convergence

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G20: Asymptotic properties
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 1007 - 1027
Copyright
Copyright © Applied Probability Trust 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.)

References

[1] Balkema, A. and de Haan, L. (1974). Residual life time at great age. Ann. Prob. 2, 792–801.Google Scholar
[2] Balkema, A. and de Haan, L. (1990). A convergence rate in extreme value theory. J. Appl. Prob. 27, 577–585.Google Scholar
[3] Beirlant, J. and Willekens, E. (1990). Rapid variation with remainder and rates of convergence. Adv. Appl. Prob. 22, 787–801.Google Scholar
[4] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
[5] Cohen, J. P. (1982). Convergence rates for the ultimate and penultimate approximations in extreme-value theory. Adv. Appl. Prob. 14, 833–854.Google Scholar
[6] De Haan, L. (1984). Slow variation and characterization of domains of attraction. In Statistical Extremes and Applications (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 131), Reidel, Dordrecht, pp. 31–48.Google Scholar
[7] De Haan, L. and Resnick, S. I. (1996). Second-order regular variation and rates of convergence in extreme-value theory. Ann. Prob. 24, 97–124.Google Scholar
[8] De Haan, L. and Stadmuller, U. (1996). Generalized regular variation of second order. J. Austral. Math. Soc. A 61, 381–395.CrossRefGoogle Scholar
[9] Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd edn. R. E. Krieger, Melbourne, FL.Google Scholar
[10] Gomes, M. I. (1984). Penultimate limiting forms in extreme value theory. Ann. Inst. Statist. Math. 36, 71–85.CrossRefGoogle Scholar
[11] Pickands, J. III (1975). Statistical inference using extreme order statistics. Ann. Statist. 3, 119–131.Google Scholar
[12] Smith, R. (1982). Uniform rate of convergence in extreme value theory. Adv. Appl. Prob. 14, 543–565.Google Scholar
[13] Worms, R. (2002). Comparaison de deux indicateurs de la régularité en variation pour les queues de distributions. C. R. Math. Acad. Sci. Paris 334, 709–712.Google Scholar