Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T14:34:44.861Z Has data issue: false hasContentIssue false
  • English
  • Français

Multivariate extremes, aggregation and dependence in elliptical distributions

Part of: Multivariate analysis Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Henrik Hult  and
Filip Lindskog
Henrik Hult*
Affiliation:
KTH, Stockholm
Filip Lindskog*
Affiliation:
ETH, Zürich
*
∗ Postal address: Department of Mathematics, KTH, S-100 44 Stockholm, Sweden.
∗∗ Postal address: RiskLab, Department of Mathematics, ETH Zentrum, CH-8092 Zürich, Switzerland. Email address: lindskog@math.ethz.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall's tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearman's rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

Keywords

Elliptical distributionmultivariate extremeregular variationtail dependenceKendall's tauSpearman's rho

MSC classification

Primary: 60E05: Distributions
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 587 - 608
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Basrak, B. (2000). The sample autocorrelation function of non-linear time series. , University of Groningen.Google Scholar
[2] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
[3] Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.Google Scholar
[4] Cambanis, S., Huang, S. and Simons, G. (1981). On the theory of elliptically contoured distributions. J. Multivar. Anal. 11, 368–385.CrossRefGoogle Scholar
[5] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
[6] Embrechts, P., McNeil, A. J. and Straumann, D. (2002). Correlation and dependence in risk management: properties and pitfalls. In Risk Management: Value at Risk and Beyond, eds Dempster, M. A. H., Cambridge University Press, pp. 176–223.Google Scholar
[7] Fang, K.-T., Kotz, S. and Ng, K.-W. (1987). Symmetric Multivariate and Related Distributions. Chapman and Hall, London.Google Scholar
[8] Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie-Verlag, Berlin.CrossRefGoogle Scholar
[9] Lindskog, F., McNeil, A. J. and Schmock, U. (2001). Kendall's tau for Elliptical distributions. Preprint, RiskLab. Available at http://www.risklab.ch/Papers/.Google Scholar
[10] Mikosch, T. (2002). Modeling dependence and tails of financial time series. To appear in SemStat: Seminaire Europeen de Statistique, Extreme Value Theory and Applications, eds Finkenstadt, B. and Rootzén, H., Chapman and Hall, London.Google Scholar
[11] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
[12] Starica, C., (1999). Multivariate extremes for models with constant conditional correlations. J. Empirical Finance 6, 515–553.Google Scholar