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Bootstrap approximation of tail dependence function

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  • Peng, Liang
  • Qi, Yongcheng
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    Abstract

    For estimating a rare event via the multivariate extreme value theory, the so-called tail dependence function has to be investigated (see [L. de Haan, J. de Ronde, Sea and wind: Multivariate extremes at work, Extremes 1 (1998) 7-45]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see [X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series, 1992] or [R. Schmidt, U. Stadtmüller, Nonparametric estimation of tail dependence, Scand. J. Stat. 33 (2006) 307-335]. In this paper, we first derive a bootstrap approximation for a tail dependence function with an approximation rate via the construction approach developed by [K. Chen, S.H. Lo, On a mapping approach to investigating the bootstrap accuracy, Probab. Theory Relat. Fields 107 (1997) 197-217], and then apply it to construct a confidence band for the tail dependence function. A simulation study is conducted to assess the accuracy of the bootstrap approach.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 99 (2008)
    Issue (Month): 8 (September)
    Pages: 1807-1824

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    Handle: RePEc:eee:jmvana:v:99:y:2008:i:8:p:1807-1824

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    Related research

    Keywords: 62G32 62G09 Bootstrap Confidence band Dependence function Extremes Tail empirical process;

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    References

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    1. Hall, Peter, 1990. "Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 177-203, February.
    2. Einmahl, J.H.J. & Haan, L.F.M. de & Li, D., 2006. "Weighted approximations of tail copula processes with applications to testing the bivariate extreme value condition," Open Access publications from Tilburg University urn:nbn:nl:ui:12-174864, Tilburg University.
    3. J. Danielsson & L. de Haan & L. Peng & C.G. de Vries, 1997. "Using a Bootstrap Method to choose the Sample Fraction in Tail Index Estimation," Tinbergen Institute Discussion Papers 97-016/4, Tinbergen Institute.
    4. Einmahl, J. & Mason, D., 1988. "Strong limit theorems for weighted quantile processes," Open Access publications from Tilburg University urn:nbn:nl:ui:12-125711, Tilburg University.
    5. Geluk, J.L. & de Haan, L.F.M., 2002. "On bootstrap sample size in extreme value theory," Econometric Institute Research Papers EI 2002-40, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    6. Drees, Holger & Huang, Xin, 1998. "Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 25-47, January.
    7. EL-NOUTY Charles & GUILLOU Armelle, 2000. "On The Bootstrap Accuracy Of The Pareto Index," Statistics & Risk Modeling, De Gruyter, vol. 18(3), pages 275-290, March.
    8. Einmahl, J. H. J. & Ruymgaart, F. H., 1987. "The almost sure behavior of the oscillation modulus of the multivariate empirical process," Statistics & Probability Letters, Elsevier, vol. 6(2), pages 87-96, November.
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    Cited by:
    1. Carsten Bormann & Melanie Schienle & and Julia Schaumburg, 2014. "A Test for the Portion of Bivariate Dependence in Multivariate Tail Risk," Tinbergen Institute Discussion Papers 14-024/III, Tinbergen Institute, revised 23 Jun 2014.
    2. Liang Peng & Yongcheng Qi, 2010. "Smoothed jackknife empirical likelihood method for tail copulas," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 19(3), pages 514-536, November.

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