TEST

, Volume 22, Issue 4, pp 549–579 | Cite as

Estimating the upcrossings index

  • J. R. Sebastião
  • A. P. Martins
  • H. Ferreira
  • L. Pereira
Original Paper

Abstract

For stationary sequences, under general dependence restrictions, any limiting point process for time normalized upcrossings of high levels is a compound Poisson process, i.e., there is a clustering of high upcrossings, where the underlying Poisson points represent cluster positions and the multiplicities correspond to cluster sizes. For such classes of stationary sequences, there exists the upcrossings index η, 0≤η≤1, which is directly related to the extremal index θ, 0≤θ≤1, for suitable high levels. In this paper, we consider the problem of estimating the upcrossings index η for a class of stationary sequences satisfying a mild oscillation restriction. For the proposed estimator, properties such as consistency and asymptotic normality are studied. Finally, the performance of the estimator is assessed through simulation studies for autoregressive processes and case studies in the fields of environment and finance. Comparisons with other estimators derived from well known estimators of the extremal index are also presented.

Keywords

Upcrossings index Local dependence conditions Consistency and asymptotic normality 

Mathematics Subject Classification (2000)

60G70 

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Copyright information

© Sociedad de Estadística e Investigación Operativa 2013

Authors and Affiliations

  • J. R. Sebastião
    • 1
  • A. P. Martins
    • 2
  • H. Ferreira
    • 2
  • L. Pereira
    • 2
  1. 1.Escola Superior de GestãoInstituto Politécnico de Castelo BrancoCastelo BrancoPortugal
  2. 2.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal

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