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Extremes

, Volume 21, Issue 4, pp 581–600 | Cite as

An estimator of the stable tail dependence function based on the empirical beta copula

  • Anna Kiriliouk
  • Johan Segers
  • Laleh Tafakori
Article
  • 90 Downloads

Abstract

The replacement of indicator functions by integrated beta kernels in the definition of the empirical tail dependence function is shown to produce a smoothed version of the latter estimator with the same asymptotic distribution but superior finite-sample performance. The link of the new estimator with the empirical beta copula enables a simple but effective resampling scheme.

Keywords

Bernstein polynomial Brown–Resnick process Bootstrap Copula Empirical process Max-linear model Tail copula Tail dependence Weak convergence 

Mathematics Subject Classification (2010)

62G32 62G30 

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Notes

Acknowledgments

A. Kiriliouk gratefully acknowledges support from the Fonds de la Recherche Scientifique (FNRS).

J. Segers gratefully acknowledges funding by contract “Projet d’Actions de Recherche Concertées” No. 12/17-045 of the “Communauté française de Belgique” and by IAP research network Grant P7/06 of the Belgian government (Belgian Science Policy).

L. Tafakori would like to thank the Australian Research Council for supporting this work through Laureate Fellowship FL130100039.

References

  1. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, New York (2004)CrossRefGoogle Scholar
  2. Beirlant, J., Escobar-Bach, M., Goegebeur, Y., Guillou, A.: Bias-corrected estimation of stable tail dependence function. J. Multivar. Anal. 143, 453–466 (2016)MathSciNetCrossRefGoogle Scholar
  3. Berghaus, B., Segers, J.: Weak convergence of the weighted empirical beta copula process. J. Multivar. Anal. arXiv:1705.06924 (2017)
  4. Berghaus, B., Bücher, A., Volgushev, S.: Weak convergence of the empirical copula process with respect to weighted metrics. Bernoulli 23(1), 743–772 (2017)MathSciNetCrossRefGoogle Scholar
  5. Bücher, A., Dette, H.: Multiplier bootstrap of tail copulas with applications. Bernoulli 19(5A), 1655–1687 (2013)MathSciNetCrossRefGoogle Scholar
  6. Bücher, A., Segers, J., Volgushev, S.: When uniform weak convergence fails: empirical processes for dependence functions and residuals via epi- and hypographs. Ann. Stat. 42(4), 1598–1634 (2014)MathSciNetCrossRefGoogle Scholar
  7. Bücher, A., Jäschke, S., Wied, D.: Nonparametric tests for constant tail dependence with an application to energy and finance. J. Econ. 187(1), 154–168 (2015)MathSciNetCrossRefGoogle Scholar
  8. Coles, S.G., Tawn, J.A.: Modelling extreme multivariate events. J. R. Stat. Soc. Ser. B (Stat Methodol.) 53(2), 377–392 (1991)MathSciNetzbMATHGoogle Scholar
  9. de Haan, L., Ferreira, A.: Extreme Value Theory: an Introduction. Springer, Berlin (2006)CrossRefGoogle Scholar
  10. de Haan, L., Resnick, S.I.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie Verwandte Geb. 40(4), 317–337 (1977)MathSciNetCrossRefGoogle Scholar
  11. Deheuvels, P.: La fonction de dépendance empirique et ses propriétés. un test non paramétrique d’indépendance. Acad. R. Belg. Bull. Cl. Sci. (5) 65(6), 274–292 (1979)zbMATHGoogle Scholar
  12. Drees, H., Huang, X.: Best attainable rates of convergence for estimators of the stable tail dependence function. J. Multivar. Anal. 64(1), 25–47 (1998)MathSciNetCrossRefGoogle Scholar
  13. Einmahl, J.H.J., de Haan, L., Li, D.: Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition. Ann. Stat. 34(4), 1987–2014 (2006)MathSciNetCrossRefGoogle Scholar
  14. Einmahl, J.H.J., Krajina, A., Segers, J.: An M-estimator for tail dependence in arbitrary dimensions. Ann. Stat. 40(3), 1764–1793 (2012)MathSciNetCrossRefGoogle Scholar
  15. Einmahl, J.H.J., Kiriliouk, A., Segers, J.: A continuous updating weighted least squares estimator of tail dependence in high dimensions. Extremes.  https://doi.org/10.1007/s10687-017-0303-7 (2017)MathSciNetCrossRefGoogle Scholar
  16. Fougères, A.-L., de Haan, L., Mercadier, C.: Bias correction in multivariate extremes. Ann. Stat. 43(2), 903–934 (2015)MathSciNetCrossRefGoogle Scholar
  17. Genton, M.G., Ma, Y., Sang, H.: On the likelihood function of Gaussian max-stable processes. Biometrika 98(2), 481–488 (2011)MathSciNetCrossRefGoogle Scholar
  18. Gissibl, N., Klüppelberg, C.: Max-linear models on directed acyclic graphs. To appear in Bernoulli arXiv:1512.07522 [math.PR] (2015)
  19. Huang, X.: Statistics of bivariate extreme values. Ph. D. thesis, Erasmus University Rotterdam, Tinbergen Institute Research Series 22 (1992)Google Scholar
  20. Huser, R., Davison, A.: Composite likelihood estimation for the Brown–Resnick process. Biometrika 100(2), 511–518 (2013)MathSciNetCrossRefGoogle Scholar
  21. Janssen, P., Swanepoel, J., Veraverbeke, N.: Large sample behavior of the bernstein copula estimator. J. Stat. Plan. Inference 142(5), 1189–1197 (2012)MathSciNetCrossRefGoogle Scholar
  22. Joe, H.: Families of min-stable multivariate exponential and multivariate extreme value distributions. Stat. Probab. Lett. 9, 75–81 (1990)MathSciNetCrossRefGoogle Scholar
  23. Kabluchko, Z., Schlather, M., de Haan, L.: Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37(5), 2042–2065 (2009)MathSciNetCrossRefGoogle Scholar
  24. Kiriliouk, A.: Hypothesis testing for tail dependence parameters on the boundary of the parameter space with application to generalized max-linear models. arXiv:1708.07019 [stat.ME]. (2017)
  25. Kojadinovic, I., Yan, J.: Modeling multivariate distributions with continuous margins using the copula R package. J. Stat. Softw. 34(9), 1–20 (2010)CrossRefGoogle Scholar
  26. Peng, L., Qi, Y.: Bootstrap approximation of tail dependence function. J. Multivar. Anal. 99, 1807–1824 (2008)MathSciNetCrossRefGoogle Scholar
  27. Pickands, J.: Multivariate extreme value distributions. In: Proceedings of the 43rd Session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981). With a discussion, vol. 49, pp 859–878, 894–902 (1981)Google Scholar
  28. R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2017)Google Scholar
  29. Ressel, P.: Homogeneous distributions–and a spectral representation of classical mean values and stable tail dependence functions. J. Multivar. Anal. 117(1), 246–256 (2013)MathSciNetCrossRefGoogle Scholar
  30. Ribatet, M: SpatialExtremes: Modelling Spatial Extremes. R package version 2.0-4 (2017)Google Scholar
  31. Sancetta, A., Satchell, S.: The bernstein copula and its applications to modeling and approximations of multivariate distributions. Econ. Theory 20(03), 535–562 (2004)MathSciNetCrossRefGoogle Scholar
  32. Schmidt, R., Stadtmüller, U.: Non-parametric estimation of tail dependence. Scand. J. Stat. 33(2), 307–335 (2006)MathSciNetCrossRefGoogle Scholar
  33. Segers, J., Sibuya, M., Tsukahara, H.: The empirical beta copula. J. Multivar. Anal. 155, 35–51 (2017)MathSciNetCrossRefGoogle Scholar
  34. Sklar, M.: Fonctions de répartition à n dimensions et leurs marges. Université Paris 8 (1959)Google Scholar
  35. Tawn, J.A.: Modelling multivariate extreme value distributions. Biometrika 77(2), 245–253 (1990)MathSciNetCrossRefGoogle Scholar
  36. van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer, New York (1996)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of EconometricsErasmus University RotterdamRotterdamThe Netherlands
  2. 2.Institut de Statistique, Biostatistique et Sciences ActuariellesUniversité Catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.School of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia

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