Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-05T04:31:01.497Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

Published online by Cambridge University Press:  01 July 2016

Jonathan P. Cohen
Jonathan P. Cohen*
Affiliation:
Imperial College, London
*
∗Present address: Department of Theoretical Statistics, University of Minnesota, 270 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Keywords

EXTREME-VALUE DISTRIBUTIONRATE OF CONVERGENCE
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 4 , December 1982 , pp. 833 - 854
Copyright
Copyright © Applied Probability Trust 1982 

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

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington D.C.Google Scholar
Anderson, C. W. (1971) Contributions to the Asymptotic Theory of Extreme Values. , University of London.Google Scholar
Cohen, J. P. (1982) The penultimate form of approximation to normal extremes. Adv. Appl. Prob. 14, 324–339.CrossRefGoogle Scholar
Fisher, R. A. and Tippett, L. H. C. (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Camb. Phil. Soc. 24, 180–190.Google Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Gomes, M. I. (1978) Some Probabilistic and Statistical Problems in Extreme Value Theory. , University of Sheffield.Google Scholar
Haan, L. De (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts 32, Amsterdam.Google Scholar
Hall, P. (1979) On the rate of convergence of normal extremes. J. Appl. Prob. 16, 433–439.Google Scholar
Hall, P. (1980) Estimating probabilities for normal extremes. Adv. Appl. Prob. 12, 491–500.CrossRefGoogle Scholar
Hall, W. J. and Wellner, J. A. (1979) The rate of convergence in law of the maximum of an exponential sample. Statistica Neerlandica 33, 151–154.Google Scholar
Smith, R. L. (1982) Uniform rates of convergence in extreme-value theory. Adv. Appl. Prob. 14, 600–622.Google Scholar
Titchmarsh, E. C. (1939) The Theory of Functions. Oxford University Press, London.Google Scholar
Von Mises, R. (1936) La distribution de la plus grande de n valeurs. Rev. Math. Union Interbalkan. 1, 141–160.Google Scholar