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Measuring Comonotonicity in M-Dimensional Vectors

Published online by Cambridge University Press:  09 August 2013

Inge Koch  and
Ann De Schepper
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Abstract

In this contribution, a new measure of comonotonicity for m-dimensional vectors is introduced, with values between zero, representing the independent situation, and one, reflecting a completely comonotonic situation. The main characteristics of this coefficient are examined, and the relations with common dependence measures are analysed. A sample-based version of the comonotonicity coefficient is also derived. Special attention is paid to the explanation of the accuracy of the convex order bound method of Goovaerts, Dhaene et al. in the case of cash flows with Gaussian discounting processes.

Keywords

Comonotonicitydependencecorrelationconcordancecopulamultivariate
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 1 , May 2011 , pp. 191 - 213
Copyright
Copyright © International Actuarial Association 2011

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References

Aas, K. (2004) Modelling the dependence structure of financial assets: a survey of four copulas. Report SAMBA/22/04 Norwegian Computing Center.Google Scholar
Cherubini, U., Luciano, E. and Vecchiato, W. (2004) Copula methods in finance. Wiley, 293 p.Google Scholar
Denneberg, D. (1994) Non-additive Measure and Integral. Kluwer Academic Publishers, Boston.Google Scholar
Denuit, M. and Scaillet, O. (2004) Non-parametric tests for positive quadrant dependence. Journal of Financial Econometrics, 2(3), 422–450.Google Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002a) The Concept of Comonotonicity in Actuarial Science and Finance: Theory. Insurance: Mathematics & Economics, 31(1), 3–34.Google Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002b) The Concept of Comonotonicity in Actuarial Science and Finance: Applications. Insurance: Mathematics & Economics, 31(2), 133–162.Google Scholar
Drouet Mari, D. and Kotz, S. (2001) Correlation and Dependence. World Scientific Publishing Company, 236 p.Google Scholar
Embrechts, P., Lindskog, F. and McNeil, A. (2003) Modelling Dependence with Copulas and Applications to Risk Management. In: Handbook of Heavy Tailed Distributions in Finance Ed: S. Rachev, Elsevier, Chapter 8, 329–384.Google Scholar
Embrechts, P., McNeil, A. and Straumann, D. (1998) Correlation and Dependence in Risk Management: Properties and Pitfalls. In: Risk management: value at risk and beyond, Cambridge University Press, 176–223.Google Scholar
Fernández, B.F. and González-Barrios, J.M. (2004) Multidimensional dependency measures. Journal of Multivariate Analysis, 89(2), 351–370.Google Scholar
Genest, C. and MacKay, J. (1986) The Joy of Copulas: Bivariate Distributions with Uniform Marginals. American Statistician, 40, 280–283.Google Scholar
Goovaerts, M.J., Dhaene, J. and De Schepper, A. (2000) Stochastic Upper Bounds for Present Value Functions. Journal of Risk and Insurance, 67(1), 1–14.Google Scholar
Hall, P., Horowitz, J.L. and Jing, B.-Y. (1995) On blocking rules for the bootstrap with dependent data. Biometrika, 82(3), 561–574.Google Scholar
Joe, H. (1990) Multivariate concordance. Journal of Multivariate Analysis, 35(1), 12–30.Google Scholar
Joe, H. (1993) Parametric Family of Multivariate Distributions with Given Margins. Journal of Multivariate Analysis, 46(2), 262–282.Google Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. Chapman & Hall, London, 399 p.Google Scholar
Kaas, R., Dhaene, J. and Goovaerts, M. (2000) Upper and Lower Bounds for Sums of Random Variables. Insurance: Mathematics & Economics, 27(2), 151–168.Google Scholar
Kaas, R., Dhaene, J., Vyncke, D., Goovaerts, M. and Denuit, M. (2002) A simple geometric proof that comonotonic risks have the convex-largest sum. Astin bulletin, 32(1), 71–80.Google Scholar
Koch, I. and De Schepper, A. (2006) The Comonotonicity Coefficient: A New Measure of Positive Dependence in a Multivariate Setting. Research Report / UA, Faculteit TEW, RPS-2006-030, available at http://www.ua.ac.be/tew.Google Scholar
Kotz, S. and Nadarajah, S. (2004) Multivariate T Distributions and Their Applications. Cambridge University Press.Google Scholar
Lehmann, E.L. (1966) Some Concepts of Dependence. The Annals of Mathematical Statistics, 37(5), 1137–1153.Google Scholar
McNeil, A., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, techniques and Tools. Princeton University Press, 608 p.Google Scholar
Melchiori, M.R. (2003) Which Archimedean copula is the right one? YieldCurve.com e-Journal, www.YieldCurve.com.Google Scholar
Micheas, A.C. and Zografos, K. (2006) Measuring stochastic dependence using ø-divergence. Journal of Multivariate Analysis, 97(3), 765–784.Google Scholar
Nelsen, R.B. (2006) An Introduction to Copulas. 2nd Edition, Springer-Verlag, New York, 216 p.Google Scholar
Pi-Erh, L. (1987) Measures of association between vectors. Communications in Statististics: Theory and Methods, 16, 321–338.Google Scholar
Scaillet, O. (2005) A Kolmogorov-Smirnov type test for positive quadrant dependence. Canadian Journal of Statistics, 33, 415–427.Google Scholar
Scarsini, M. (1984) On measures of concordance. Stochastica, 8, 201–218.Google Scholar
Schmidt, R. (2002) Tail Dependence for Elliptically Contoured Distributions. Mathematical Methods of Operations Research, 55, 301–327.Google Scholar
Schweizer, B. and Wolff, E.F. (1981) On Nonparametric Measures of Dependence for Random Variables. The Annals of Statistics, 9(4), 879–885.Google Scholar
Tjostheim, D. (1996) Measures of dependence and tests of independence. Statistics, 28, 249–284.Google Scholar
Wolff, E.F. (1980) N-dimensional Measures of Dependence. Stochastica, 4(3), 175–188.Google Scholar
Zografos, K. (2000) Measures of multivariate dependence based on a distance between Fisher information matrices. Journal of Statistical Planning and Inference, 89, 91–107.Google Scholar