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A note on generalized Pareto distributions and thek upper extremes
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  • Published: December 1990

A note on generalized Pareto distributions and thek upper extremes

  • Michael Falk1 

Probability Theory and Related Fields volume 85, pages 499–503 (1990)Cite this article

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Summary

Denote byD(n, k) the maximum distance between the distribution function of thekth-largest order statistic in aniid sample of sizen, equally standardized ink, and of its corresponding limiting extreme value distribution.

Suppose that the underlying distribution function is ultimately continuous and strictly increasing. It is shown in the present paper thatD(n,k) converges to zero for any sequencek=k (n) with\(\mathop {\lim }\limits_{n \to \infty } k/n = 0\) if and only if the underlying distribution is ultimately a generalized Pareto distribution. ThusgPds do not only yield the best rate of joint convergence of extremes, but they are also the only distributions where there actually is convergence.

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Authors and Affiliations

  1. Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt, Ostenstrasse 26-28, D-8078, Eichstät, Federal Republic of Germany

    Michael Falk

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  1. Michael Falk
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Falk, M. A note on generalized Pareto distributions and thek upper extremes. Probab. Th. Rel. Fields 85, 499–503 (1990). https://doi.org/10.1007/BF01203167

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  • Received: 18 May 1989

  • Revised: 29 November 1989

  • Issue Date: December 1990

  • DOI: https://doi.org/10.1007/BF01203167

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Keywords

  • Distribution Function
  • Stochastic Process
  • Probability Theory
  • Order Statistic
  • Mathematical Biology
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