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Bivariate copula decomposition in terms of comonotonicity, countermonotonicity and independence. (English) Zbl 1098.62070
Summary: Copulas are statistical tools for modelling the multivariate dependence structure among variables in a distribution free way. This paper investigates bivariate copula structure; the existence and uniqueness of a bivariate copula decomposition into a comonotonic, an independent, a countermonotonic, and an indecomposable part are proved, while the coefficients are determined from partial derivatives of the corresponding copula. Moreover, for the indecomposable part, an optimal convex approximation is provided and analyzed on the basis of the usual criterion. Some applications of the decomposition in finance and insurance are mentioned.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G99 Nonparametric inference
62P05 Applications of statistics to actuarial sciences and financial mathematics
62H20 Measures of association (correlation, canonical correlation, etc.)
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[1] Alink, S.; Löwe, M.; Wüthrich, M.V., Diversification of aggregate dependent risks, Insurance: mathematics and economics, 35, 77-95, (2004) · Zbl 1052.62105
[2] Chow, Y.S.; Teicher, H., Probability theory, (1988), Springer-Verlag New York
[3] Denneberg, D., Non-additive measure and integral, (1994), Kluwer Academic Publishers Boston · Zbl 0826.28002
[4] Dhaene, J.; Denuit, M.; Goovaerts, M.J.; Kaas, R.; Vynche, D., The concept of comonotonicity in actuarial science and finance: theory, Insurance: mathematics and economics, 31, 1, 3-33, (2002) · Zbl 1051.62107
[5] Dhaene, J.; Denuit, M.; Goovaerts, M.J.; Kaas, R.; Vynche, D., The concept of comonotonicity in actuarial science and finance: application, Insurance: mathematics and economics, 31, 2, 133-161, (2002) · Zbl 1037.62107
[6] Embrechets, P.; Lindskog, F.; McNeil, A., Modelling dependence with copulas and applications to risk management, (2001)
[7] Frees, E.W.; Valdez, E.A., Understanding relationship using copulas, North American actuarial journal, 2, 1, 1-24, (1996)
[8] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[9] Jondeau, E., Rockinger, M., 2002. Conditional dependence of financial series: The copula-Garch model. In: FAME-International Center for Financial Asset Management and Engineering, No. 69 · Zbl 1043.62110
[10] Kass, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern actuarial risk theory, (2001), Kluwer Academic Publishers Boston
[11] Lojasiewicz, S., An introduction to the theory of real functions, (1988), John Wiley & Sons New York
[12] Mari, D.D.; Kotz, S., Correlation and dependence, (2001), Imperial College Press Singapore · Zbl 0977.62004
[13] Mikusinski, P.; Sherwood, H.; Taylor, M.D., Probabilistic interpretations of copulas and their convex sums, (), 95-112 · Zbl 0733.60023
[14] Muller, A.; Scarsini, M., Stochastic comparison of random vectors with a common copula, Mathematics of operations research, 26, 4, 723-740, (2001) · Zbl 1082.60504
[15] Nelsen, R.B., Copulas and associations, (), 51-74
[16] Nelsen, R.B., An introduction to copulas, (1999), Springer-Verlag New York · Zbl 0909.62052
[17] Sarathy, R.; Muralidhar, K.; Parsa, R., Perturbing nonnormal confidential attributes: the copula approach, Management science, 48, 12, 1613-1627, (2002)
[18] Schmeidler, D., Integral representation without additivity, Proceedings of the American mathematical society, 97, 255-261, (1986) · Zbl 0687.28008
[19] Schweizer, B., Thirty years of copulas, (), 13-55 · Zbl 0727.60001
[20] Schweizer, B.; Wolff, E.F., On non-parametric measure of dependence for random variables, Annals of statistics, 9, 870-885, (1981) · Zbl 0468.62012
[21] Smith, M.D., Modelling sample selection using Archimedean copulas, Econometrics journal, 6, 99-123, (2003) · Zbl 1037.62047
[22] Wang, S.S.; Young, V.R.; Panjer, H.H., Axiomatic characterization of insurance prices, Insurance: mathematics and economics, 21, 173-183, (1997) · Zbl 0959.62099
[23] Yaari, M.E., The dual theory of choice under risk, Econometrica, 55, 95-114, (1987) · Zbl 0616.90005
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