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On the Tail Behavior of Sums of Dependent Risks

Published online by Cambridge University Press:  17 April 2015

Philippe Barbe ,
Anne-Laure Fougères  and
Christian Genest
Philippe Barbe
Affiliation:
Centre national de la recherche scientifique, 90, rue de Vaugirard, 75006 Paris, France
Anne-Laure Fougères
Affiliation:
Équipe Modal’X Unité de formation et de recherche SEGMI, Université Paris X – Nanterre, 200, avenue de la République, 92000 Nanterre, France
Christian Genest
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec (Québec), Canada G1K 7P4
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Abstract

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The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

Keywords

Archimedean copulaextremal behaviormultivariate regular variationPickands’ representation
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 36 , Issue 2 , November 2006 , pp. 361 - 373
Copyright
Copyright © ASTIN Bulletin 2006

References

Alink, S., Löwe, M. and Wüthrich, M.V. (2004) Diversification of aggregate dependent risks. Insurance: Mathematics and Economics 35, 77–95.Google Scholar
Alink, S., Löwe, M. and Wüthrich, M.V. (2005) Analysis of the expected shortfall of aggregate dependent risks. Astin Bulletin 35, 25–43.CrossRefGoogle Scholar
Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004) Statistics of extremes: theory and applications. Wiley, New York.CrossRefGoogle Scholar
Capéraà, P., Fougères, A.-L. and Genest, C. (2000) Bivariate distributions with given extreme value attractor. Journal of Multivariate Analysis 72, 30–49.CrossRefGoogle Scholar
Cherubini, U., Luciano, E. and Vecchiato, W. (2004) Copula methods in finance. Wiley, New York.CrossRefGoogle Scholar
COLES, S.G. and TAWN, J.A. (1991) Modelling extreme multivariate events. Journal of the Royal Statistical Society Series B 53, 377–392.Google Scholar
De Haan, L. and Resnick, S.I. (1977) Limit theory for multidimensional sample extremes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 40, 317–337.CrossRefGoogle Scholar
Feller, W. (1971) An introduction to probability theory and its applications. Volume II, Second Edition, Wiley, New York.Google Scholar
Frees, E.W. and Valdez, E.A. (1998) Understanding relationships using copulas. North American Actuarial Journal 2, 1–25.CrossRefGoogle Scholar
Genest, C. and Mackay, R.J. (1986) Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. The Canadian Journal of Statistics 14, 145–159.CrossRefGoogle Scholar
Genest, C. and Rivest, L.-P. (1989) A characterization of Gumbel’s family of extreme value distributions. Statistics and Probability Letters 8, 207–211.CrossRefGoogle Scholar
Huang, X. (1992) Statistics of bivariate extremes. Doctoral dissertation, Tinbergen Institute Research series no 22, Erasmus Universiteit, Rotterdam, The Netherlands.Google Scholar
Joe, H. (1997) Multivariate models and dependence concepts. Chapman & Hall, London.Google Scholar
Mikosch, T. (2004) Modeling dependence and tails of financial time series. In Extreme Values in Finance, Telecommunications, and the Environment, (Finkelstädt, B. and Rootzén, H., Eds.), Chapman & Hall, London, pp. 185–286.Google Scholar
Nelsen, R.B. (1999) An introduction to copulas. Springer, New York.CrossRefGoogle Scholar
Pickands, J. (1981) Multivariate extreme value distributions. Proceedings, 43rd Session of the International Statistical Institute. Buenos Aires, Argentina, Book 2, pp. 859–878.Google Scholar
Resnick, S.I. (1987) Extreme values, regular variation, and point processes. Springer, New York.Google Scholar
Resnick, S.I. (2004) The extremal dependence measure and asymptotic independence. Stochastic Models 20, 205–227.CrossRefGoogle Scholar
Tawn, J.A. (1988) Bivariate extreme value theory: Models and estimation. Biometrika 75, 397–415.CrossRefGoogle Scholar
Wüthrich, M.V. (2003) Asymptotic value-at-risk estimates for sums of dependent random variables. Astin Bulletin 33, 75–92.CrossRefGoogle Scholar