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Exceedances of Levels and kth Largest Maxima

  • M. R. Leadbetter
  • Georg Lindgren
  • Holger Rootzén
Part of the Springer Series in Statistics book series (SSS)

Abstract

In this chapter, we investigate properties of the exceedances of levels {u n } by ξi, ξ2,…, i.e. the points i for which ξi, > un, and as consequences, obtain limiting distributional results for the kth largest value among ξ1,…, ξ1. In particular, when the familiar assumption \(n\left( {1 - F\left( {u_n } \right)} \right) \to \tau \left( {0 < \tau < \infty } \right)\) holds (Equation (1.5.1)), it will be shown that the exceedances take on a Poisson character as n becomes large. This leads to the limiting distributions for the kth largest values for any fixed rank k = 1, 2,… (the kth “extreme order statistics”) and to their limiting joint distributions.

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Copyright information

© Springer-Verlag New York Inc. 1983

Authors and Affiliations

  • M. R. Leadbetter
    • 1
  • Georg Lindgren
    • 2
  • Holger Rootzén
    • 3
  1. 1.Department of StatisticsThe University of North CarolinaChapel HillUSA
  2. 2.Department of Mathematical StatisticsUniversity of LundLundSweden
  3. 3.Institute of Mathematical StatisticsUniversity of CopenhagenCopenhagen øDenmark

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