Advances in Applied Probability

Extremal behavior of Archimedean copulas

Martin Larsson and Johanna Nešlehová

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We show how the extremal behavior of d-variate Archimedean copulas can be deduced from their stochastic representation as the survival dependence structure of an l1-symmetric distribution (see McNeil and Nešlehová (2009)). We show that the extremal behavior of the radial part of the representation is determined by its Williamson d-transform. This leads in turn to simple proofs and extensions of recent results characterizing the domain of attraction of Archimedean copulas, their upper and lower tail-dependence indices, as well as their associated threshold copulas. We outline some of the practical implications of their results for the construction of Archimedean models with specific tail behavior and give counterexamples of Archimedean copulas whose coefficient of lower tail dependence does not exist.

Article information

Adv. in Appl. Probab., Volume 43, Number 1 (2011), 195-216.

First available in Project Euclid: 15 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory

Archimedean copula domain of attraction extreme value copula ell_1-norm symmetric distribution regular variation simplex distribution tail dependence threshold copula Williamson transform


Larsson, Martin; Nešlehová, Johanna. Extremal behavior of Archimedean copulas. Adv. in Appl. Probab. 43 (2011), no. 1, 195--216. doi:10.1239/aap/1300198519.

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  • Ballerini, R. (1994). Archimedean copulas, exchangeability, and max-stability. J. Appl. Prob. 31, 383–390.
  • Barbe, P., Fougères, A.-L. and Genest, C. (2006). On the tail behavior of sums of dependent risks. ASTIN Bull. 36, 361–373.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.
  • Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Teor. Verojat. Primen. 10, 351–360.
  • Capéraà, P., Fougères, A.-L. and Genest, C. (2000). Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72, 30–49.
  • Charpentier, A. and Juri, A. (2004). Limiting dependence structure for credit default. Working paper 2004-16.
  • Charpentier, A. and Segers, J. (2007). Lower tail dependence for Archimedean copulas: characterizations and pitfalls. Insurance Math. Econom. 40, 525–532.
  • Charpentier, A. and Segers, J. (2009). Tails of multivariate Archimedean copulas. J. Multivariate Anal. 100, 1521–1537.
  • Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151.
  • Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.
  • Coles, S., Heffernan, J. and Tawn, J. (2000). Dependence measures for extreme value analyses. Extremes 2, 339–365.
  • De Haan, L. and Ferreira, A. (2006). Extreme Value Theory. Springer, New York.
  • Demarta, S. (2007). The copula approach to modeling multivariate extreme values. Doctoral Thesis, ETH Zürich.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.
  • Falk, M. and Reiss, R.-D. (2005). On Pickands coordinates in arbitrary dimensions. J. Multivariate Anal. 92, 426–453.
  • Frees, E. W. and Valdez, E. A. (1998). Understanding relationships using copulas. N. Amer. Actuarial J. 2, 1–25.
  • Genest, C. and Rivest, L.-P. (1989). A characterization of Gumbel's family of extreme value distributions. Statist. Prob. Lett. 8, 207–211.
  • Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Prob. 34, 587–608.
  • Joe, H. (1993). Multivariate dependence measures and data analysis. Comput. Statist. Data Anal. 16, 279–297.
  • Juri, A. and Wüthrich, M. V. (2002). Copula convergence theorems for tail events. Insurance Math. Econom. 30, 405–420.
  • Juri, A. and Wüthrich, M. V. (2003). Tail dependence from a distributional point of view. Extremes 6, 213–246.
  • Larsson, M. (2008). Tail properties of multivariate Archimedean copulas. Masters Thesis, ETH Zürich.
  • Marshall, A. W. and Olkin, I. (1988). Families of multivariate distributions. J. Amer. Statist. Assoc. 83, 834–841.
  • McNeil, A. J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions. Ann. Statist. 37, 3059–3097.
  • McNeil, A. J. and Nešlehová, J. (2010). From Archimedean to Liouville copulas. J. Multivariate Anal. 101, 1772–1790.
  • McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management. Princeton University Press.
  • Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.
  • Oakes, D. (1994). Multivariate survival distributions. J. Nonparametric Statist. 3, 343–354.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.
  • Schönbucher, P. (2003). Credit Derivatives Pricing Models. John Wiley, Chichester.
  • Schweizer, B. and Sklar, A. (1983). Probabilistic Metric Spaces. North-Holland Publishing, New York.
  • Zhang, D., Wells, M. T. and Peng, L. (2008). Nonparametric estimation of the dependence function for a multivariate extreme value distribution. J. Multivariate Anal. 99, 577–588.