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Estimating the upcrossings index

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  • J. Sebastião
  • A. Martins
  • H. Ferreira
  • L. Pereira

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. Copyright Sociedad de Estadística e Investigación Operativa 2013

Suggested Citation

  • J. Sebastião & A. Martins & H. Ferreira & L. Pereira, 2013. "Estimating the upcrossings index," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(4), pages 549-579, November.
    • Handle: RePEc:spr:testjl:v:22:y:2013:i:4:p:549-579
    DOI: 10.1007/s11749-013-0315-9
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    References listed on IDEAS

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    1. Hsing, Tailen, 1991. "Estimating the parameters of rare events," Stochastic Processes and their Applications, Elsevier, vol. 37(1), pages 117-139, February.
    2. F. Laurini & J. A. Tawn, 2006. "The extremal index for GARCH(1,1) processes with t-distributed innovations," Economics Department Working Papers 2006-SE01, Department of Economics, Parma University (Italy).
    3. Marta Ferreira & Helena Ferreira, 2012. "On extremal dependence: some contributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(3), pages 566-583, September.
    4. Christopher A. T. Ferro & Johan Segers, 2003. "Inference for clusters of extreme values," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 545-556, May.
    5. Gomes, M. Ivette & Hall, Andreia & Miranda, M. Cristina, 2008. "Subsampling techniques and the Jackknife methodology in the estimation of the extremal index," Computational Statistics & Data Analysis, Elsevier, vol. 52(4), pages 2022-2041, January.
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    Cited by:

    1. Marta Ferreira, 2024. "Extremal index: estimation and resampling," Computational Statistics, Springer, vol. 39(5), pages 2703-2720, July.
    2. A. P. Martins & J. R. Sebastião, 2019. "Methods for estimating the upcrossings index: improvements and comparison," Statistical Papers, Springer, vol. 60(4), pages 1317-1347, August.

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