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On extremal dependence: some contributions

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  • Marta Ferreira
  • Helena Ferreira

Abstract

The usual coefficients of tail dependence are based on exceedances of high values. These extremal events are useful and widely used in literature but an adverse situation may also occur with the upcrossing of a high level. In this context we define upcrossings-tail dependence coefficients and analyze all types of dependence coming out. We will prove that these coefficients are related to multivariate tail dependence coefficients already known in literature. We shall see that the upcrossings-tail dependence coefficients have the interesting feature of congregating both “temporal” and “spatial” dependence. The coefficients of tail dependence can also be applied to stationary sequences and hence measure the tail dependence in time. Results concerning connections with the extremal index and the upcrossings index as well as with local dependence conditions will be stated. Several illustrative examples will be exploited and a small note on inference will be given by presenting estimators derived from the stated results and respective properties. Copyright Sociedad de Estadística e Investigación Operativa 2012

Suggested Citation

  • 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.
    • Handle: RePEc:spr:testjl:v:21:y:2012:i:3:p:566-583
    DOI: 10.1007/s11749-011-0261-3
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    References listed on IDEAS

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    1. Rafael Schmidt & Ulrich Stadtmüller, 2006. "Non‐parametric Estimation of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 307-335, June.
    2. Frahm, Gabriel & Junker, Markus & Schmidt, Rafael, 2005. "Estimating the tail-dependence coefficient: Properties and pitfalls," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 80-100, August.
    3. Anthony W. Ledford & Jonathan A. Tawn, 1997. "Modelling Dependence within Joint Tail Regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 475-499.
    4. Peng, L., 1999. "Estimation of the coefficient of tail dependence in bivariate extremes," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 399-409, July.
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    Cited by:

    1. 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.
    2. Ferreira, Helena & Ferreira, Marta, 2014. "Extremal behavior of pMAX processes," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 46-57.
    3. 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.
    4. Helena Ferreira & Marta Ferreira, 2021. "Tail dependence and smoothness of time series," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 198-210, March.
    5. Ferreira Helena & Ferreira Marta, 2022. "The stopped clock model," Dependence Modeling, De Gruyter, vol. 10(1), pages 48-57, January.

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