Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T16:42:02.247Z Has data issue: false hasContentIssue false
  • English
  • Français

Rapid variation with remainder and rates of convergence

Published online by Cambridge University Press:  01 July 2016

J. Beirlant  and
E. Willekens
J. Beirlant*
Affiliation:
Katholieke Universiteit Leuven
E. Willekens*
Affiliation:
University of Technology, Eindhoven
*
∗Postal address: Faculteit Wetenschappen, Katholieke Universiteit Leuven, Dept. Wiskunde, Celestijinenlaan 200 B, B-3030 Leuven, Belgium.
∗∗Postal address: Department of Mathematics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

Keywords

REGULAR VARIATIONEXTREME VALUE THEORYHILL ESTIMATOR
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 4 , December 1990 , pp. 787 - 801
Copyright
Copyright © Applied Probability Trust 1990 

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

Anderson, C. W. (1971) Contributions to the Asymptotic Theory of Extreme Values. Ph.D. Thesis, University of London.Google Scholar
Balkema, A. A. and De Haan, L. (1990) A convergence rate in extreme-value theory. J. Appl. Prob. 27, 577–585.CrossRefGoogle Scholar
Beirlant, J. and Teugels, J. L. (1987) Asymptotics of Hill's estimator. Theory Prob. Appl. 31, 463–469.CrossRefGoogle Scholar
Bingham, N. H. and Goldie, C. M. (1983) On one-sided Tauberian theorems. Analysis 3, 159–188.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Encyclopedia Math. Appl. 27, Cambridge University Press.Google Scholar
Cohen, J. P. (1982) Convergence rates for the ultimate and penultimate approximations in extreme-value theory. Adv. Appl. Prob. 14, 833–854.Google Scholar
Dekkers, A. and De Haan, L. (1989) On the estimation of the extreme-value index and large quantile estimation. Ann. Statist. 17, 1795–1832.Google Scholar
De Haan, L. (1970) On Regular Variations and its Application to the Weak Convergence of Sample Extremes. Math. Centre Tract 32, Amsterdam.Google Scholar
De Haan, L. (1974) Equivalence classes of regularly varying functions. Stoch. Proc. Appl. 2, 243–259.CrossRefGoogle Scholar
Falk, M (1985) Uniform convergence of extreme order statistics. Habilitationsthesis, University of Siegen.Google Scholar
Goldie, C. M. and Smith, R. L. (1987) Slow variation with remainder: theory and applications. Quart. J. Math. Oxford 38, 45–71.Google Scholar
Hill, B. M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163–1174.Google Scholar
Omey, E. and Rachev, S. (1989) On the rate of convergence in extreme value theory. Theory Prob. Appl. 33, 560–565.CrossRefGoogle Scholar
Omey, E. and Willekens, E. (1988) p-variation with remainder. J. London Math. Soc. (2) 37, 105–118.Google Scholar
Reiss, R. D. (1987) Estimating the tail index of the claim size distribution. Blatter DGVM 1, 21–25.CrossRefGoogle Scholar
Resnick, S. I. (1986) Uniform rates of convergence to extreme value distributions. In Probability and Statistics: Essays in honor of F. A. Gray bill , ed. Srivastava, J., North-Holland, Amsterdam.Google Scholar
Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Smith, R. L. (1982) Uniform rates of convergence in extreme-value theory. Adv. Appl. Prob. 14, 600–622.Google Scholar