On an interaction function for copulas

Abstract

We study properties of a local dependence function of Wang for copulas. In this paper this dependence function is called the mixed derivative measure of interactions as it is a mixed derivative of a log of a density function. It is stressed that this measure is not margin free in the sense that the interaction function of a density and the corresponding copula are not equal. We show that there is no Archimedean copula with constant interactions. The interaction function is positive (negative) for an Archimedean copula density whose second derivative of the generator is log convex (log concave). Moreover, the only Archimedean copula with interactions proportional to its density is Frank’s copula. We obtain some preliminary results concerning the connection between the behaviour of the interaction function and the tail dependence of the distribution. Moreover, the notion of an interaction function has been extended to more than two dimensional case, and we study its properties for a canonical Archimedean copula.