On bivariate Kumaraswamy-distorted copulas

Abstract

We propose families of bivariate copulas based on the Kumaraswamy distortion of existing copulas. With the additional two parameters in the Kumaraswamy distribution, the induced copulas permit more flexibility in tail behaviors. The framework employed in this paper also provides a method for generating new Archimedean copula models. Two theorems linking the original tail dependence behaviors and those of the distorted copula are derived for distortions that are asymptotically proportional to the power transformation in the lower tail and to the dual-power transformation in the upper tail. We also derive explicit formulas for the Kendall’s τ coefficients, tail order parameters and tail order functions for the induced copulas when Gumbel, Clayton, Frank and Galambos are distorted. An empirical application is also presented.