A Procedure of Perturbation Leading to a New Class of Asymmetric Copulas

We propose and analyze a new class of copulas (generated by two univariate functions) that exhibit asymmetric nonlinear dependence, resulting from an antisymmetric perturbation of the independence copula. Within this framework, we construct a parametric subclass of asymmetric copulas for which we derive the analytical expression of Kendall’s tau (τ). We present properties related to asymmetry and dependence measures and provide several examples. This novel feature of our class represents a significant advancement in the context of the most studied perturbed copulas, which have recently been the focus of numerous scientific investigations. In particular, we introduce Liebscher-type constructions and FGM perturbations, resulting in a large general framework for perturbation theory of copulas using elementary mathematical analysis. In addition, we illustrate statements with examples and figures.