×

Tail dependence for multivariate copulas and its monotonicity. (English) Zbl 1152.62342

Summary: The tail dependence indexes of a multivariate distribution describe the amount of dependence in the upper right tail or lower left tail of the distribution and can be used to analyse the dependence among extremal random events. This paper examines the tail dependence of multivariate \(t\)-distributions whose copulas are not explicitly accessible. The tractable formulas of tail dependence indexes of a multivariate \(t\)-distribution are derived in terms of the joint moments of its underlying multivariate normal distribution, and the monotonicity properties of these indexes with respect to the distribution parameters are established. Simulation results are presented to illustrate the results.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Buhl, C., Reich, C., Wegmann, P., 2002. Extremal dependence between return risk and liquidity risk: An analysis for the Swiss Market. Technical Report. Department of Finance. University of Basel
[2] Embrechts, P.; Lindskog, F.; McNeil, A., Modeling dependence with copulas and applications to risk management, (), 329-384, (Chapter 8)
[3] Frahm, G., On the extreme dependence coefficient of multivariate distributions, Probability & statistics letter, 76, 1470-1481, (2006) · Zbl 1120.62035
[4] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[5] Kole, E., Koedijk, K., Verbeek, M., 2006. Selecting copulas for risk management. Technical Report. Erasmus School of Economics and Business Economics. Erasmus University Rotterdam, The Netherlands
[6] Lauritzen, S.L., Graphical models, (1996), Oxford Science Publications New York · Zbl 0907.62001
[7] Li, H., 2006a. Tail dependence comparison of survival Marshall-Olkin copulas. Technical Report 2006-4. Department of Mathematics. Washington State University, Pullman, WA; Methodology and Computing in Applied Probability, August 2007 (published online) (in press) (http://www.math.wsu.edu/TRS/2006-index.html) · Zbl 1142.62035
[8] Li, H., 2006b. Tail dependence of multivariate Pareto distributions. Technical Report 2006-6. Department of Mathematics. Washington State University, Pullman, WA 99164 (http://www.math.wsu.edu/TRS/2006-index.html)
[9] Nelsen, R., An introduction to copulas, (1999), Springer New York · Zbl 0909.62052
[10] Resnick, S., Extreme values, regularly variation, and point processes, (1987), Springer New York
[11] Robert, C.P.; Casella, G., Monte Carlo statistical methods, (1999), Springer New York · Zbl 0935.62005
[12] Schmidt, R., Tail dependence for elliptically contoured distributions, Mathematical methods of operations research, 55, 301-327, (2002) · Zbl 1015.62052
[13] Sklar, A., Fonctions de répartition à \(n\) dimensions et leurs marges, (1959), Publications de l’Institut de statistique de l’Université de Paris
[14] Tong, Y.L., Probability inequalities in multivariate distributions, vol. 8, (1980), Academic Press New York, pp. 229-231 · Zbl 0455.60003
[15] Venter, G.G., 2003. Quantifying correlated reinsurance exposures with copulas, In: Casualty Actuarial Society Forum, Spring, pp. 215-229
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.