A family of transformed copulas with singular component

In this paper, we present a family of bivariate copulas by transforming a given copula function with 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 the transformed copulas are obtained. Finally, some properties of the transformed copula are discussed, such as the totally positive of order 2 and the concordance order.