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The Pareto Copula, Aggregation of Risks, and the Emperor's Socks

Part of: Nonparametric inference Distribution theory - Probability Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Claudia Klüppelberg  and
Sidney I. Resnick
Claudia Klüppelberg*
Affiliation:
Munich University of Technology
Sidney I. Resnick*
Affiliation:
Cornell University
*
∗Postal address: Chair of Mathematical Statistics, Center for Mathematical Sciences, Munich University of Technology, Boltzmannstrasse 3, 85747 Garching bei München, Germany. Email address: [email protected]
∗∗Postal address: School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: [email protected]
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Abstract

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The copula of a multivariate distribution is the distribution transformed so that one-dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions, and we call the resulting distribution the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima, and empirical processes. We discuss implications for aggregation of risk and offer some examples.

Keywords

Regular variationriskmaximal domain of attractioncopulaPareto

MSC classification

Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes 60G70: Extreme value theory; extremal processes 60F17: Functional limit theorems; invariance principles 62G30: Order statistics; empirical distribution functions
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 67 - 84
Copyright
Copyright © Applied Probability Trust 2008 

References

Albrecher, H., Asmussen, S. and Kortschak, D. (2006). Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes 9, 107–130.Google Scholar
Alink, S., Löwe, M. and Wüthrich, M. V. (2004). Diversification of aggregate dependent risks. Insurance Math. Econom. 35, 77–95.CrossRefGoogle Scholar
Barndorff-Nielsen, O. and Lindner, A. (2006). Lévy copulas: dynamics and transforms of upsilon type. Scand. J. Statist. 34, 298–316.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Böcker, K. and Klüppelberg, C. (2006). Multivariate models for operational risk. Submitted.Google Scholar
Bregman, Y. and Klüppelberg, C. (2005). Ruin estimation in multivariate models with Clayton dependence structure. Scand. Actuarial J. 2005, 462–480.Google Scholar
Cont, R. and Tankov, P. (2004). Financial Modelling With Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
De Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematisch Centrum, Amsterdam.Google Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.Google Scholar
De Haan, L. and Resnick, S. I. (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40, 317–337.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extreme Events for Insurance and Finance. Springer, Berlin.Google Scholar
Geluk, J. L. and de Haan, L. (1987). Regular Variation, Extensions and Tauberian Theorems (CWI Tract. 40). CWI, Amsterdam.Google Scholar
Kallsen, J. and Tankov, P. (2006). Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivariate Anal. 97, 1551–1572.Google Scholar
Maulik, K., Resnick, S. I. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699.Google Scholar
Mikosch, T. (2005). How to model multivariate extremes if one must? Statist. Neerlandica 59, 324–338.CrossRefGoogle Scholar
Mikosch, T. (2006). Copulas: Tales and facts. Extremes 9, 3–20.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.Google Scholar
Resnick, S. I. (2007). Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.Google Scholar