A Mixture of Clayton, Gumbel, and Frank Copulas: A Complete Dependence Model

Abstract

Knowledge of the dependence between random variables is necessary in the area of risk assessment and evaluation. Some of the existing Archimedean copulas, namely the Clayton and the Gumbel copulas, allow for higher correlations on the extreme left and right, respectively. In this study, we use the idea of convex combinations to build a hybrid Clayton–Gumbel–Frank copula that provides all dependence scenarios from existing Archimedean copulas. The corresponding density and conditional distribution functions of the derived models for two random variables, as well as an estimator for the proportion parameter associated with the proposed model, are also derived. The results show that the proposed model is able to show any case of dependence by providing coefficients for the upper tail and lower tail dependence.