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Estimating the spectral measure of an extreme value distribution

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  • Einmahl, J.H.J

    (Tilburg University)

  • Haan, L. de
  • Sinha, A.

Abstract

Let (X1, Y1), (X2, Y2),..., (Xn, Yn) be a random sample from a bivariate distribution function F which is in the domain of attraction of a bivariate extreme value distribution function G. This G is characterized by the extreme value indices and its spectral measure or angular measure. The extreme value indices determine both the marginals and the spectral measure determines the dependence structure. In this paper, we construct an empirical measure, based on the sample, which is a consistent estimator of the spectral measure. We also show for positive extreme value indices the asymptotic normality of the estimator under a suitable 2nd order strengthening of the bivariate domain of attraction condition.

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Bibliographic Info

Paper provided by Tilburg University in its series Open Access publications from Tilburg University with number urn:nbn:nl:ui:12-125733.

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Date of creation: 1997
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Publication status: Published in Stochastic Processes and their Applications (1997) v.70, p.143-171
Handle: RePEc:ner:tilbur:urn:nbn:nl:ui:12-125733

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Web page: http://www.tilburguniversity.edu/

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References

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  1. Einmahl, J.H.J. & Haan, L. de & Xin, H., 1993. "Estimating a multidimensional extreme-value distribution," Open Access publications from Tilburg University, Tilburg University urn:nbn:nl:ui:12-142070, Tilburg University.
  2. Deheuvels, Paul, 1991. "On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions," Statistics & Probability Letters, Elsevier, Elsevier, vol. 12(5), pages 429-439, November.
  3. Deheuvels, Paul & Tiago de Oliveira, José, 1989. "On the non-parametric estimation of the bivariate extreme-value distributions," Statistics & Probability Letters, Elsevier, Elsevier, vol. 8(4), pages 315-323, September.
  4. Einmahl, John H. J., 1997. "Poisson and Gaussian approximation of weighted local empirical processes," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 70(1), pages 31-58, October.
  5. 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, Tilburg University urn:nbn:nl:ui:12-125712, Tilburg University.
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Cited by:
  1. 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, Elsevier, vol. 99(4), pages 577-588, April.
  2. Tim Bollerslev & Viktor Todorov, 2010. "Jump Tails, Extreme Dependencies, and the Distribution of Stock Returns," CREATES Research Papers 2010-64, School of Economics and Management, University of Aarhus.
  3. 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.
  4. Falk, Michael & Reiss, Rolf-Dieter, 2005. "On Pickands coordinates in arbitrary dimensions," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 92(2), pages 426-453, February.
  5. Falk, Michael & Reiss, Rolf-Dieter, 2005. "On the distribution of Pickands coordinates in bivariate EV and GP models," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 93(2), pages 267-295, April.
  6. Einmahl, J.H.J., 1997. "Poisson and Gaussian approximation of weighted local empirical processes," Open Access publications from Tilburg University, Tilburg University urn:nbn:nl:ui:12-125732, Tilburg University.
  7. Capéraà, Philippe & Fougères, Anne-Laure & Genest, Christian, 2000. "Bivariate Distributions with Given Extreme Value Attractor," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 72(1), pages 30-49, January.
  8. Georg Mainik & Ludger Rüschendorf, 2010. "On optimal portfolio diversification with respect to extreme risks," Finance and Stochastics, Springer, Springer, vol. 14(4), pages 593-623, December.

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