On a conjecture about the Frank copula family

Authors:
B. De Baets;H. De Meyer
Affiliations:
Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent, Belgium;Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, B-9000 Gent, Belgium
Venue:
Fuzzy Sets and Systems
Year:
2013

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Abstract

It is shown that any member of the Frank copula family fulfills a family of functional inequalities which arises as a family of sufficient conditions that allow to characterize the transitivity of reciprocal relations expressing the winning probabilities among random variables artificially coupled by a same copula. The inequalities are parameterized by a single integer parameter. The first of these inequalities expresses symmetry of the copula, while the second one expresses an upper bound on its volume. Subsequent inequalities are increasingly restrictive.