A class of bivariate independence copula transformations

This paper deals with the problem of characterizing all functions f : [0, 1] -> R+ such that C-f (x, y) = xyf ((1 - x)(1 - y)), x, y is an element of [0, 1] is a bivariate copula. We provide a complete characterization for the two cases: (i) when C-f is, in addition, totally positive of order 2 (TP2) and (ii) when f is twice continuously differentiable. In general, the function f need only be twice differentiable Lebesgue almost everywhere as shown by investigating necessary conditions for C-f to be a copula. The paper also contains numerous examples illustrating obtained results and connections to known facts from the literature. Moreover, several properties of such copulas are described. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

This paper provides a comprehensive characterization of functions f: [0, 1] -> R+ such that C-f satisfies specific properties, including two special cases and general conditions. Various examples are given to illustrate the obtained results, and properties of such copulas are described in detail.