Functional inequalities characterizing the frank family of copulas

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. © 2010 Springer-Verlag Berlin Heidelberg.