Miltivariate measures of dependence based on copulas

• Authors: Heilpern Stanisław
• Languages of publication: EN

Abstracts

EN

The paper is devoted to the multivariate measures of dependence. In contrast to the classical approach, where the pairs of variables are studied, we investigate the dependence of more than two variables. We mainly consider the measures based on copulas. These are the multivariable generalizations of the known coefficients of such correlation as Spearman’s rho, Kendall’s tau, Blomquist’s beta and Gini’s gamma. We present the definitions, the constructions and the basic properties of such multivariate measures of dependence. The case of large number of dimension, greater than two, presents more complications. We have several different versions of such generalization in this case and the lower bound of the values of such measures of dependence are close to zero. We also study the multivariate tail dependences. The last part of the paper is devoted to the estimation of multivariable versions of Spearman’s rho coefficient.

Keywords

EN

multivariate measures of dependence; copulas; tail dependences; estimation

• Publisher: Wydawnictwo Uniwersytetu Ekonomicznego we Wrocławiu
• Journal: Mathematical Economics
• Year: 2014
• Issue: 10(17)
• Pages: 17-32

Contributors

author

Heilpern Stanisław

References

• Blomqvist N. (1950). On a measure of dependence between two random variables. Ann. Math. Stat. 21(4). Pp. 593-600.
• Dolati A., Ubeda-Flores M. (2006). On measures of multivariate concordance. J. Prob. Stat. Sci. 4(2). Pp. 147-164.
• Domański Cz. (1990). Testy statystyczne. PWE. Warszawa.
• Embrechts P., Kluppelberg C., Mikosch T. (1997). Modelling External Events for Insurance and Finance. Springer. Berlin.
• Embrechts P., Lindskog F., McNeil A. (2001). Modelling Dependence with Copulas and Applications to Risk Management. ETH Zurich, preprint.
• Embrechts P., McNeil A., Straumann D. (2002). Correlation and dependency in risk management: properties and pitfalls. In: M.A.H. Dempster (ed.). Risk Management: Value at Risk and Beyond. Cambridge University Press. Cambridge. Pp. 176-223.
• Frahm G. (2006). On the extremal dependence coefficient of multivariate distributions. Stat. Probab. Lett. 76(14). Pp. 1470-1481.
• Genest C., MacKay R.J. (1986). The joy of copulas: Bivariate distributions with uniform marginals. Am. Stat. 40(4). Pp. 280-285.
• Joe H. (1990). Multivariate concordance. J. Multivar. Anal. 35(1). Pp. 12-30.
• Nelsen R.B. (1996). Nonparametric measures of multivariate association. In: Distribution with Fixed Marginals and Related Topics. IMS Lecture Notes – Monograph Series 28. Institite of Mathematical Statistics. Hayward. Pp. 223- 232.
• Nelsen R.B. (2006). An Introduction to Copulas (2nd edition). Springer. New York.
• Schmid F., Schmidt R. (2007). Multivariate extensions of Spearman’s rho and related statistics. Stat. Probab. Lett. 77(4). Pp. 407-416.
• Schmid F., Schmidt R., Blumentritt T., Gaißer S., Ruppert M. (2010). Copula-Based Measures of Multivariate Association. In: P. Jaworski, F. Durante, W. Hardle, T. Rychlik (ed.). Copula Theory and Its Applications. 10. Springer. Berlin.
• Sibuya M. (1960). Bivariate extreme statistics. Ann. Inst. Math. 11(3). Pp. 195- 210.
• Taylor M.D. (2007). Multivariate measures of concordance. Ann. Inst. Stat. Math. 59(4). Pp. 789-806.
• Wolff E.F. (1980). N-dimensional measures of dependence. Stochastica 4(3). Pp. 175-188.