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Nonparametric estimation of the dependence function for a multivariate extreme value distribution

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  • Zhang, Dabao
  • Wells, Martin T.
  • Peng, Liang

Abstract

Understanding and modeling dependence structures for multivariate extreme values are of interest in a number of application areas. One of the well-known approaches is to investigate the Pickands dependence function. In the bivariate setting, there exist several estimators for estimating the Pickands dependence function which assume known marginal distributions [J. Pickands, Multivariate extreme value distributions, Bull. Internat. Statist. Inst., 49 (1981) 859-878; P. Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 (1991) 429-439; P. Hall, N. Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (2000) 835-844; P. Capéraà, A.-L. Fougères, C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (1997) 567-577]. In this paper, we generalize the bivariate results to p-variate multivariate extreme value distributions with p[greater-or-equal, slanted]2. We demonstrate that the proposed estimators are consistent and asymptotically normal as well as have excellent small sample behavior.

Suggested Citation

  • Zhang, Dabao & Wells, Martin T. & Peng, Liang, 2008. "Nonparametric estimation of the dependence function for a multivariate extreme value distribution," Journal of Multivariate Analysis, Elsevier, vol. 99(4), pages 577-588, April.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:4:p:577-588
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    References listed on IDEAS

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    1. Deheuvels, Paul, 1991. "On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 429-439, November.
    2. Einmahl, John H.J. & de Haan, Laurens & Sinha, Ashoke Kumar, 1997. "Estimating the spectral measure of an extreme value distribution," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 143-171, October.
    3. Einmahl, J.H.J. & de Haan, L.F.M. & Piterbarg, V.I., 2001. "Nonparametric estimation of the spectral measure of an extreme value distribution," Other publications TiSEM c3485b9b-a0bd-456f-9baa-0, Tilburg University, School of Economics and Management.
    4. Falk, Michael & Reiss, Rolf-Dieter, 2005. "On Pickands coordinates in arbitrary dimensions," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 426-453, February.
    5. de Oliveira, J. Tiago, 1989. "Intrinsic estimation of the dependence structure for bivariate extremes," Statistics & Probability Letters, Elsevier, vol. 8(3), pages 213-218, August.
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    Cited by:

    1. Gudendorf, Gordon & Segers, Johan, 2011. "Nonparametric estimation of multivariate extreme-value copulas," LIDAM Discussion Papers ISBA 2011018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Papastathopoulos, Ioannis & Tawn, Jonathan A., 2014. "Dependence properties of multivariate max-stable distributions," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 134-140.
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    4. Gudendorf, Gordon & Segers, Johan, 2011. "Nonparametric estimation of an extreme-value copula in arbitrary dimensions," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 37-47, January.
    5. Dutfoy Anne & Parey Sylvie & Roche Nicolas, 2014. "Multivariate Extreme Value Theory - A Tutorial with Applications to Hydrology and Meteorology," Dependence Modeling, De Gruyter, vol. 2(1), pages 1-19, June.
    6. Enkelejd Hashorva & Simone A. Padoan & Stefano Rizzelli, 2021. "Multivariate extremes over a random number of observations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(3), pages 845-880, September.
    7. Gardes, Laurent & Girard, Stéphane, 2015. "Nonparametric estimation of the conditional tail copula," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 1-16.
    8. Marcon, Giulia & Padoan, Simone & Naveau, Philippe & Muliere, Pietro & Segers, Johan, 2016. "Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials," LIDAM Discussion Papers ISBA 2016020, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Guillou, Armelle & Padoan, Simone A. & Rizzelli, Stefano, 2018. "Inference for asymptotically independent samples of extremes," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 114-135.

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