On (a,b)-transformations of conjunctive functions

Abstract

We introduce and study transformations that assign to each conjunctive real-valued function F:[0,1]2→R and any pair of parameters (a,b)∈[0,1]2 a function Fa,b:[0,1]2→R that is shown to be conjunctive, too. In more details, we study the proposed transformations in the classes of all binary copulas, quasi-copulas, conjunctive k-Lipschitz functions and the class of all conjunctive Lipschitz functions. We discuss the invariance of these classes with respect to all possible transformations. An interesting result is, that although the class of all copulas is invariant with respect to all considered transformations, the only copula that is not changed by any of them is the product copula. We also show that (a,b)-transformations of copulas yield the same results as constructions of copulas based on special measure-preserving transformations; and using measure-preserving transformations we extend the studied binary constructions to n-dimensional copulas. Finally, we focus our attention on consecutive realization of the proposed transformations in the class of binary copulas and we also show several properties of (a,b)-transforms of the basic copulas, including some dependence parameters.