On Two Simple, Three-Parameter, Three-Dimensional, Non-Exchangeable Copulas

Abstract

In this paper, we use the notion of a copula to provide theoretical contributions to the development of three-dimensional dependence models. In particular, we suggest two new three-dimensional copulas whose forms are simple and adaptable; they are based on polynomials, power functions, and three tuning parameters. In order to rely on the existing literature, we mention that the second copula can be viewed as a generalization of the three-dimensional Farlie- Gumbel-Morgenstern copula. Both copulas have the feature of being non-exchangeable (for most of the parameter values). Theoretical results are demonstrated, including wide admissible sets of values for the parameters and closed-form expressions for the medial correlation and Spearman’s rho. By using our methodology, the limitations imposed by the exchangeable property, which are typical of traditional three-dimensional copulas in the literature, are thus overcome, and new approaches to dependence modeling are opened up.