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Extremal behavior of Archimedean copulas

Part of: Limit theorems Distribution theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Martin Larsson  and
Johanna Nešlehová
Martin Larsson*
Affiliation:
Cornell University
Johanna Nešlehová*
Affiliation:
McGill University
*
∗ Postal address: Department of Operations Research and Information Engineering, Cornell University, 206 Rhodes Hall, Ithaca, NY 14853, USA.
∗∗ Postal address: Department of Mathematics and Statistics, McGill University, 805, rue Sherbrooke Ouest, Montréal (Québec), H3A 2K6, Canada. Email address: [email protected]
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Abstract

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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.

Keywords

Archimedean copuladomain of attractionextreme value copulaℓ1-norm symmetric distributionregular variationsimplex distributiontail dependencethreshold copulaWilliamson transform

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 195 - 216
Copyright
Copyright © Applied Probability Trust 2011 

References

Ballerini, R. (1994). Archimedean copulas, exchangeability, and max-stability. J. Appl. Prob. 31, 383–390.CrossRefGoogle Scholar
Barbe, P., Fougères, A.-L. and Genest, C. (2006). On the tail behavior of sums of dependent risks. ASTIN Bull. 36, 361–373.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Teor. Verojat. Primen. 10, 351–360.Google Scholar
Capéraà, P., Fougères, A.-L. and Genest, C. (2000). Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72, 30–49.CrossRefGoogle Scholar
Charpentier, A. and Juri, A. (2004). Limiting dependence structure for credit default. Working paper 2004-16.Google Scholar
Charpentier, A. and Segers, J. (2007). Lower tail dependence for Archimedean copulas: characterizations and pitfalls. Insurance Math. Econom. 40, 525–532.CrossRefGoogle Scholar
Charpentier, A. and Segers, J. (2009). Tails of multivariate Archimedean copulas. J. Multivariate Anal. 100, 1521–1537.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.CrossRefGoogle Scholar
Coles, S., Heffernan, J. and Tawn, J. (2000). Dependence measures for extreme value analyses. Extremes 2, 339–365.CrossRefGoogle Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory. Springer, New York.CrossRefGoogle Scholar
Demarta, S. (2007). The copula approach to modeling multivariate extreme values. , ETH Zürich.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.CrossRefGoogle Scholar
Falk, M. and Reiss, R.-D. (2005). On Pickands coordinates in arbitrary dimensions. J. Multivariate Anal. 92, 426–453.CrossRefGoogle Scholar
Frees, E. W. and Valdez, E. A. (1998). Understanding relationships using copulas. N. Amer. Actuarial J. 2, 1–25.CrossRefGoogle Scholar
Genest, C. and Rivest, L.-P. (1989). A characterization of Gumbel's family of extreme value distributions. Statist. Prob. Lett. 8, 207–211.CrossRefGoogle Scholar
Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Prob. 34, 587–608.CrossRefGoogle Scholar
Joe, H. (1993). Multivariate dependence measures and data analysis. Comput. Statist. Data Anal. 16, 279–297.CrossRefGoogle Scholar
Juri, A. and Wüthrich, M. V. (2002). Copula convergence theorems for tail events. Insurance Math. Econom. 30, 405–420.CrossRefGoogle Scholar
Juri, A. and Wüthrich, M. V. (2003). Tail dependence from a distributional point of view. Extremes 6, 213–246.CrossRefGoogle Scholar
Larsson, M. (2008). Tail properties of multivariate Archimedean copulas. , ETH Zärich.Google Scholar
Marshall, A. W. and Olkin, I. (1988). Families of multivariate distributions. J. Amer. Statist. Assoc. 83, 834–841.CrossRefGoogle Scholar
McNeil, A. J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and ℓ1-norm symmetric distributions. Ann. Statist. 37, 3059–3097.CrossRefGoogle Scholar
McNeil, A. J. and Nešlehová, J. (2010). From Archimedean to Liouville copulas. J. Multivariate Anal. 101, 1772–1790.CrossRefGoogle Scholar
McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management. Princeton University Press.Google Scholar
Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.Google Scholar
Oakes, D. (1994). Multivariate survival distributions. J. Nonparametric Statist. 3, 343–354.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.Google Scholar
Schönbucher, P. (2003). Credit Derivatives Pricing Models. John Wiley, Chichester.Google Scholar
Schweizer, B. and Sklar, A. (1983). Probabilistic Metric Spaces. North-Holland Publishing, New York.Google Scholar
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.CrossRefGoogle Scholar