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Nonparametric estimation of an extreme-value copula in arbitrary dimensions

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  • Gudendorf, Gordon
  • Segers, Johan

Abstract

Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an i.i.d. random sample from a multivariate distribution with known margins and an unknown extreme-value copula, an extension of the Capéraà-Fougères-Genest estimator was introduced by D. Zhang, M. T. Wells and L. Peng [Nonparametric estimation of the dependence function for a multivariate extreme-value distribution, Journal of Multivariate Analysis 99 (4) (2008) 577-588]. The joint asymptotic distribution of the estimator as a random function on the simplex was not provided. Moreover, implementation of the estimator requires the choice of a number of weight functions on the simplex, the issue of their optimal selection being left unresolved. A new, simplified representation of the CFG-estimator combined with standard empirical process theory provides the means to uncover its asymptotic distribution in the space of continuous, real-valued functions on the simplex. Moreover, the ordinary least-squares estimator of the intercept in a certain linear regression model provides an adaptive version of the CFG-estimator whose asymptotic behavior is the same as if the variance-minimizing weight functions were used. As illustrated in a simulation study, the gain in efficiency can be quite sizable.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:102:y:2011:i:1:p:37-47
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    References listed on IDEAS

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    1. Obretenov, A., 1991. "On the dependence function of Sibuya in multivariate extreme value theory," Journal of Multivariate Analysis, Elsevier, vol. 36(1), pages 35-43, January.
    2. Einmahl, J.H.J. & Segers, J.J.J., 2008. "Maximum Empirical Likelihood Estimation of the Spectral Measure of an Extreme Value Distribution," Other publications TiSEM e9340b9a-fe69-4e77-8594-8, Tilburg University, School of Economics and Management.
    3. 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.
    4. 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.
    5. Einmahl, J.H.J. & Segers, J.J.J., 2009. "Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution," Other publications TiSEM ffef2e15-c4a8-471f-b730-1, Tilburg University, School of Economics and Management.
    6. Einmahl, J.H.J. & Segers, J.J.J., 2008. "Maximum Empirical Likelihood Estimation of the Spectral Measure of an Extreme Value Distribution," Other publications TiSEM e9340b9a-fe69-4e77-8594-8, Tilburg University, School of Economics and Management.
    7. Jiménez, Javier Rojo & Villa-Diharce, Enrique & Flores, Miguel, 2001. "Nonparametric Estimation of the Dependence Function in Bivariate Extreme Value Distributions," Journal of Multivariate Analysis, Elsevier, vol. 76(2), pages 159-191, February.
    8. Vaart,A. W. van der, 1998. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521496032.
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    Cited by:

    1. Mhalla, Linda & Chavez-Demoulin, Valérie & Naveau, Philippe, 2017. "Non-linear models for extremal dependence," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 49-66.
    2. Gardes, Laurent & Girard, Stéphane, 2015. "Nonparametric estimation of the conditional tail copula," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 1-16.
    3. 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).
    4. 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.
    5. 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).
    6. Bucher, Axel & Segers, Johan, 2013. "Extreme value copula estimation based on block maxima of a multivariate stationary time series," LIDAM Discussion Papers ISBA 2013049, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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