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Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function

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  • Drees, Holger
  • Huang, Xin
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    Abstract

    It is well known that a bivariate distribution belongs to the domain of attraction of an extreme value distribution G if and only if the marginals belong to the domain of attraction of the univariate marginal extreme value distributions and the dependence function converges to the stable tail dependence function of G. Hall and Welsh (1984,Ann. Statist.12, 1079-1084) and Drees (1997b,Ann. Statist., to appear) addressed the problem of finding optimal rates of convergence for estimators of the extreme value index of an univariate distribution. The present paper deals with the corresponding problem for the stable tail dependence function. First an upper bound on the rate of convergence for estimators of the stable tail dependence function is established. Then it is shown that this bound is sharp by proving that it is attained by the tail empirical dependence function. Finally, we determine the limit distribution of this estimator if the dependence function satisfies a certain second-order condition.

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    Bibliographic Info

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

    Volume (Year): 64 (1998)
    Issue (Month): 1 (January)
    Pages: 25-47

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    Handle: RePEc:eee:jmvana:v:64:y:1998:i:1:p:25-47

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    Related research

    Keywords: asymptotic normality; bivariate extreme value distribution; domain of attraction; rate of convergence; stable tail dependence function; tail empirical dependence function;

    References

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    1. Einmahl, J. & Dekkers, A. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Open Access publications from Tilburg University urn:nbn:nl:ui:12-125712, Tilburg University.
    2. Einmahl, J. H. J. & Dehaan, L. & Huang, X., 1993. "Estimating a Multidimensional Extreme-Value Distribution," Journal of Multivariate Analysis, Elsevier, vol. 47(1), pages 35-47, October.
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    Cited by:
    1. Peng, Liang & Qi, Yongcheng, 2008. "Bootstrap approximation of tail dependence function," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1807-1824, September.
    2. Falk, Michael & Reiss, Rolf Dieter, 2002. "A characterization of the rate of convergence in bivariate extreme value models," Statistics & Probability Letters, Elsevier, vol. 59(4), pages 341-351, October.
    3. Di Bernardino, Elena & Maume-Deschamps, Véronique & Prieur, Clémentine, 2013. "Estimating a bivariate tail: A copula based approach," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 81-100.
    4. Einmahl, J.H.J. & Haan, L.F.M. de & Li, D., 2004. "Weighted Approximations of Tail Copula Processes with Application to Testing the Multivariate Extreme Value Condition," Discussion Paper 2004-71, Tilburg University, Center for Economic Research.
    5. Jäschke, Stefan, 2014. "Estimation of risk measures in energy portfolios using modern copula techniques," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 359-376.
    6. Cai, J. & Einmahl, J.H.J. & Haan, L.F.M. de & Zhou, C., 2012. "Estimation of the Marginal Expected Shortfall: The Mean when a Related Variable is Extreme," Discussion Paper 2012-080, Tilburg University, Center for Economic Research.

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