Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-27T20:36:46.669Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail Dependence for Heavy-Tailed Scale Mixtures of Multivariate Distributions

Part of: Multivariate analysis Applications

Published online by Cambridge University Press:  14 July 2016

Haijun Li  and
Yannan Sun
Haijun Li*
Affiliation:
Washington State University
Yannan Sun*
Affiliation:
Washington State University
*
∗Postal address: Department of Mathematics, Washington State University, Pullman, WA 99164, USA.
∗Postal address: Department of Mathematics, Washington State University, Pullman, WA 99164, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The tail dependence of multivariate distributions is frequently studied via the tool of copulas. In this paper we develop a general method, which is based on multivariate regular variation, to evaluate the tail dependence of heavy-tailed scale mixtures of multivariate distributions, whose copulas are not explicitly accessible. Tractable formulae for tail dependence parameters are derived, and a sufficient condition under which the parameters are monotone with respect to the heavy tail index is obtained. The multivariate elliptical distributions are discussed to illustrate the results.

Keywords

Tail dependenceheavy tailmultivariate regular variationcopulahazard rate orderingmultivariate elliptical distribution

MSC classification

Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 925 - 937
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Research supported in part by NSF grant CMMI 0825960.

References

[1] Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908–920.Google Scholar
[2] Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.Google Scholar
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[4] Buhl, C., Reich, C. and Wegmann, P. (2002). Extremal dependence between return risk and liquidity risk: an analysis for the Swiss Market. Tech. Rep., Department of Tech. Rep.Google Scholar
[5] Embrechts, P., Lindskog, F. and McNeil, A. (2003). Modeling dependence with copulas and applications to risk management. In Handbook of Heavy Tailed Distributions in Finance, ed. Rachev, S., Elsevier, pp. 329–384.Google Scholar
[6] Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. Chapman and Hall, London.Google Scholar
[7] Finkenstädt, B. and Rootzén, H. (2004). Extreme Values in Finance, Telecommunications, and the Environment. Chapman and Hall/CRC, New York.Google Scholar
[8] Frahm, G. (2006). On the extreme dependence coefficient of multivariate distributions. Statist. Prob. Lett. 76, 1470–1481.CrossRefGoogle Scholar
[9] Hartmann, P., Straetmans, S., and de Vries, C. (2006). Banking system stability. A cross-Atlantic perspective. In The Risks of Financial Institutions, University of Chicago Press, pp. 133–192.Google Scholar
[10] Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Prob. 34, 587–608.Google Scholar
[11] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.Google Scholar
[12] Joe, H., Li, H. and Nikoloulopoulos, A. K. (2009). Tail dependence functions and vine copulas. To appear in J. Multivariate Anal.Google Scholar
[13] Klüppelberg, C., Kuhn, G. and Peng, L. (2008). Semi-parametric models for the multivariate tail dependence function—the asymptotically dependent. Scand. J. Statist. 35, 701–718.CrossRefGoogle Scholar
[14] Li, H. (2008). Tail dependence comparison of survival Marshall–Olkin copulas. Methodology Comput. Appl. Prob. 10, 39–54.Google Scholar
[15] Li, H. (2009). Orthant tail dependence of multivariate extreme value distributions. J. Multivariate Anal. 100, 243–256.Google Scholar
[16] McNeil, A. J., Frey, R., Embrechts, P. (2005). Quantitative Risk Management. Princeton University Press.Google Scholar
[17] Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.Google Scholar
[18] Nikoloulopoulos, A. K., Joe, H. and Li, H. (2009). Extreme value properties of multivariate t copulas. Extremes 12, 129–148.CrossRefGoogle Scholar
[19] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
[20] Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.Google Scholar
[21] Schmidt, R. (2002). Tail dependence for elliptically contoured distributions. Math. Meth. Operat. Res. 55, 301–327.Google Scholar
[22] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
[23] Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231.Google Scholar