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Multivariate Extreme Value Theory and D-Norms pp 135–160Cite as

Copulas & Multivariate Extremes

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Abstract

This chapter reveals the crucial role that copulas play in MEVT. The D-norm approach again proves to be quite a helpful tool. In particular, it turns out that a multivariate df F is in the domain of attraction of a multivariate EVD iff this is true for the univariate margins of F together with the condition that the copula of F in its upper tail is close to that of a generalized Pareto copula. As a consequence, MEVT actually means extreme value theory for copulas.

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References

  • Aulbach, S., Bayer, V., and Falk, M. (2012a). A multivariate piecing-together approach with an application to operational loss data. Bernoulli 18, 455–475. doi:10.3150/10-BEJ343.

    Article  MathSciNet  Google Scholar 

  • Aulbach, S., Falk, M., and Hofmann, M. (2012b). The multivariate piecing-together approach revisited. J. Multivariate Anal. 110, 161–170. doi:10.1016/j.jmva.2012.02.002.

    Article  MathSciNet  Google Scholar 

  • Billingsley, P. (1999). Convergence of Probability Measures. Wiley Series in Probability and Statistics, 2nd ed. Wiley, New York. doi:10.1002/9780470316962.

    MATH  Google Scholar 

  • Charpentier, A., and Segers, J. (2009). Tails of multivariate Archimedean copulas. J. Multivariate Anal. 100, 1521–1537. doi:10.1016/j.jmva.2008.12.015.

    Article  MathSciNet  Google Scholar 

  • Deheuvels, P. (1984). Probabilistic aspects of multivariate extremes. In Statistical Extremes and Applications (J. Tiago de Oliveira, ed.), 117–130. D. Reidel, Dordrecht. doi:10.1007/978-94-017-3069-3 9.

    Google Scholar 

  • Falk, M., and Guillou, A. (2008). Peaks-over-threshold stability of multivariate generalized Pareto distributions. J. Multivariate Anal. 99, 715–734. doi:10.1016/j.jmva.2007.03.009.

    Article  MathSciNet  Google Scholar 

  • Falk, M., Hüsler, J., and Reiss, R.-D. (2011). Laws of Small Numbers: Extremes and Rare Events. 3rd ed. Birkhäuser, Basel. doi:10.1007/978-3-0348-7791-6.

    Book  Google Scholar 

  • Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics. 2nd ed. Krieger, Malabar.

    MATH  Google Scholar 

  • Genest, C., and Neslehová̌, J. (2012). Copula modeling for extremes. In Encyclopedia of Environmetrics (A. El-Shaarawi and W. Piegorsch, eds.), vol. 2, 530–541. Wiley, Chichester. doi:10.1002/9780470057339.vnn018.

    Google Scholar 

  • Huang, X. (1991). Statistics of Bivariate Extreme Values. Ph.D. thesis, Tinbergen Institute Research Series.

    Google Scholar 

  • Kortschak, D., and Albrecher, H. (2009). Asymptotic results for the sum of dependent non-identically distributed random variables. Methodol. Comput. Appl. Probab. 11, 279–306. doi:10.1007/s11009-007-9053-3.

    Article  MathSciNet  Google Scholar 

  • McNeil, A. J., and Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and ℓ 1-norm symmetric distributions. Ann. Statist. 37, 3059–3097. doi:10.1214/07-AOS556.

    Article  MathSciNet  Google Scholar 

  • Nelsen, R. B. (2006). An Introduction to Copulas. Springer Series in Statistics, 2nd ed. Springer, New York. doi:10.1007/0-387-28678-0.

    Google Scholar 

  • Reiss, R.-D. (1989). Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. Springer Series in Statistics. Springer, New York. doi:10.1007/978-1-4613-9620-8.

    Book  Google Scholar 

  • Reiss, R.-D., and Thomas, M. (2007). Statistical Analysis of Extreme Values: with Applications to Insurance, Finance, Hydrology and Other Fields. 3rd ed. Birkhäuser, Basel. doi:10.1007/978-3-7643-7399-3.

    Google Scholar 

  • Rootzén, H., and Tajvidi, N. (2006). Multivariate generalized Pareto distributions. Bernoulli 12, 917–930. doi:10.3150/bj/1161614952.

    Article  MathSciNet  Google Scholar 

  • Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Pub. Inst. Stat. Univ. Paris 8, 229–231.

    MATH  Google Scholar 

  • Sklar, A. (1996). Random variables, distribution functions, and copulas – a personal look backward and forward. In Distributions with fixed marginals and related topics (L. Rüschendorf, B. Schweizer, and M. D. Taylor, eds.), Lecture Notes – Monograph Series, vol. 28, 1–14. Institute of Mathematical Statistics, Hayward, CA. doi:10.1214/lnms/1215452606.

    Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik und Informatik, Universität Würzburg, Würzburg, Germany

    Michael Falk

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  1. Michael Falk
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Falk, M. (2019). Copulas & Multivariate Extremes. In: Multivariate Extreme Value Theory and D-Norms. Springer Series in Operations Research and Financial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-03819-9_3

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