Abstract
Given a random vector with components that are pairwisely coupled by means of a same commutative copula C, we analyze the transitivity of the reciprocal relation obtained from the pairwise comparison of these components. The transitivity of this reciprocal relation can be elegantly described within the cycle-transitivity framework if the commutative copula C satisfies a countably infinite family of (functional) inequalities. Each functional inequality uniquely characterizes the Frank family of copulas. Finally, we highlight the transitivity results for a random vector whose coupling structure is captured by an extended Frank m-copula.
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References
Alsina, C.: Some functional equations in the space of uniform distribution functions. Aequationes Math. 22, 153–164 (1981)
Calvo, T., De Baets, B., Fodor, J.: The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets Syst. 120, 385–394 (2001)
De Baets, B., De Meyer, H.: Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets Syst. 152, 249–270 (2005)
De Baets, B., De Meyer, H.: On the cycle-transitive comparison of artificially coupled random variables. Internat. J. Approx. Reason. 47, 306–322 (2008)
De Baets, B., De Meyer, H., De Schuymer, B., Jenei, S.: Cyclic evaluation of transitivity of reciprocal relations. Soc. Choice Welf. 26, 217–238 (2006)
De Meyer, H., De Baets, B.: The Frank inequality (in preparation, 2010)
De Meyer, H., De Baets, B., De Schuymer, B.: On the transitivity of the comonotonic and countermonotonic comparison of random variables. J. Multivariate Anal. 98, 177–193 (2007)
De Schuymer, B., De Meyer, H., De Baets, B.: Cycle-transitive comparison of independent random variables. J. Multivariate Anal. 96, 352–373 (2005)
De Schuymer, B., De Meyer, H., De Baets, B.: Extreme copulas and the comparison of ordered lists. Theory and Decision 62, 195–217 (2007)
De Schuymer, B., De Meyer, H., De Baets, B., Jenei, S.: On the cycle-transitivity of the dice model. Theory and Decision 54, 261–285 (2003)
Frank, M.: On the simultaneous associativity of F(x,y) and x + y − F(x,y). Aequationes Math. 19, 194–226 (1979)
Genest, C.: Frank’s family of bivariate distributions. Biometrika 74, 549–555 (1987)
Klement, E., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic—Studia Logica Library, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)
Klement, E., Mesiar, R., Pap, E.: Invariant copulas. Kybernetika 38, 275–285 (2002)
Nelsen, R.: An Introduction to Copulas, 2nd edn. Springer, Heidelberg (2006)
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De Meyer, H., De Baets, B. (2010). Functional Inequalities Characterizing the Frank Family of Copulas. In: Borgelt, C., et al. Combining Soft Computing and Statistical Methods in Data Analysis. Advances in Intelligent and Soft Computing, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14746-3_21
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DOI: https://doi.org/10.1007/978-3-642-14746-3_21
Publisher Name: Springer, Berlin, Heidelberg
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