Geodesic
Bivariate copulas: Transformations, asymmetry and measures of concordance
Kybernetika, Tome 50 (2014) no. 1, pp. 109-125
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The present paper introduces a group of transformations on the collection of all bivariate copulas. This group contains an involution which is particularly useful since it provides (1) a criterion under which a given symmetric copula can be transformed into an asymmetric one and (2) a condition under which for a given copula the value of every measure of concordance is equal to zero. The group also contains a subgroup which is of particular interest since its four elements preserve symmetry, the order between two copulas and the value of every measure of concordance.
The present paper introduces a group of transformations on the collection of all bivariate copulas. This group contains an involution which is particularly useful since it provides (1) a criterion under which a given symmetric copula can be transformed into an asymmetric one and (2) a condition under which for a given copula the value of every measure of concordance is equal to zero. The group also contains a subgroup which is of particular interest since its four elements preserve symmetry, the order between two copulas and the value of every measure of concordance.
DOI : 10.14736/kyb-2014-1-0109
Classification : 20C99, 60B15, 60E05, 62E10, 62H20
Keywords: bivariate copulas; transformations; asymmetry; measures of concordance 
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Fuchs, Sebastian; Schmidt, Klaus D. Bivariate copulas: Transformations, asymmetry and measures of concordance. Kybernetika, Tome 50 (2014) no. 1, pp. 109-125. doi: 10.14736/kyb-2014-1-0109

[1] Baets, B. De, Meyer, H. De, Mesiar, R.: Asymmetric semilinear copulas. Kybernetika 43 (2007), 221-233. | MR | Zbl

[2] Durante, F., Papini, P. L.: Componentwise concave copulas and their asymmetry. Kybernetika 45 (2009), 1003-1011. | MR | Zbl

[3] Durante, F., Sempi, C.: Copula theory - an introduction. In: Copula Theory and Its Applications (P. Jaworski, F. Durante, W. K. Härdle and T. Rychlik, eds.), Springer, Berlin, Heidelberg 2010, pp. 3-31. | MR

[4] Edwards, H. H., Mikusiński, P., Taylor, M. D.: Measures of concordance determined by $D_4$-invariant copulas. Internat. J. Math. Sci. 70 (2004), 3867-3875. | DOI | MR | Zbl

[5] Fuchs, S.: Multivariate copulas - Transformations, symmetry, order and measures of concordance. Submitted for publication (2013).

[6] Klement, E. P., Mesiar, R., Pap, E.: Invariant copulas. Kybernetika 38 (2002), 275-285. | MR | Zbl

[7] Liebscher, E.: Construction of asymmetric multivariate copulas. J. Multivariate Anal. 99 (2008), 2234-2250. | DOI | MR | Zbl

[8] Nelsen, R. B.: An Introduction to Copulas. Second edition. Springer, New York 2006. | MR

[9] Nelsen, R. B.: Extremes of nonexchangeability. Statist. Papers 48 (2007), 329-336. | DOI | MR | Zbl

[10] Scarsini, M.: On measures of concordance. Stochastica 8 (1984), 201-218. | MR | Zbl

[11] Taylor, M. D.: Multivariate measures of concordance. Ann. Inst. Statist. Math. 59 (2007), 789-806. | DOI | MR | Zbl

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