A Collection of Two-Dimensional Copulas Based on an Original Parametric Ratio Scheme

Abstract

The creation of two-dimensional copulas is crucial for the proposal of novel families of two-dimensional distributions and the analysis of original dependence structures between two quantitative variables. Such copulas can be developed in a variety of ways. In this article, we provide theoretical contributions to this subject; we emphasize a new parametric ratio scheme to create copulas of the following form: C(x,y)=(b+1)xy/[b+ϕ(x,y)], where b is a constant and ϕ(x,y) is a two-dimensional function. As a notable feature, this form can operate an original trade-off between the product copula and more versatile copulas (not symmetric, with tail dependence, etc.). Instead of a global study, we examine seven concrete examples of such copulas, which have never been considered before. Most of them are extended versions of existing non-ratio copulas, such as the Celebioglu–Cuadras, Ali-Mikhail-Haq, and Gumbel–Barnett copulas. We discuss their attractive properties, including their symmetry, dominance, dependence, and correlation features. Some graphics and tables are given as complementary works. Our findings expand the horizons of new two-dimensional distributional or dependence modeling.

On the Ratio-Type Family of Copulas

“…For a brief overview of works related to ratio-type copulas, one may refer to [1,3,[16][17][18][19] and the citations therein. B 0 coincides with the independence copula, Π(u, v) = uv, whereas, for ϕ(u, v) = (1 − u)(1 − v) and θ ∈ [−1, 1], B θ corresponds to the Ali-Mikhail-Haq family of copula with parameter θ.…”