A family of transformed copulas with a singular component

Abstract In this paper, we present a family of bivariate copulas obtained by transforming a given copula function by means of two increasing functions, named as transformed copula. One distinctive characteristic of the transformed copula is its singular component along the main diagonal. Conditions guaranteeing the transformed function to be a copula function are provided, and several classes of the transformed copulas are given. The singular component along the main diagonal of the transformed copula is verified, and the tail dependence coefficients of transformed copulas are obtained. Some properties of the transformed copula are discussed, such as the total positivity of order 2 and the concordance order. Finally, conditions for the proposed transformation being invariant are given, sufficient conditions guaranteeing that the iterative transformation is a copula function are provided, and the convergence of the iterative transformation is proved under some conditions.