Theoretical Advancements on a Few New Dependence Models Based on Copulas with an Original Ratio Form

Abstract

Copulas are well-known tools for describing the relationship between two or more quantitative variables. They have recently received a lot of attention, owing to the variable dependence complexity that appears in heterogeneous modern problems. In this paper, we offer five new copulas based on a common original ratio form. All of them are defined with a single tuning parameter, and all reduce to the independence copula when this parameter is equal to zero. Wide admissible domains for this parameter are established, and the mathematical developments primarily rely on non-trivial limits, two-dimensional differentiations, suitable factorizations, and mathematical inequalities. The corresponding functions and characteristics of the proposed copulas are looked at in some important details. In particular, as common features, it is shown that they are diagonally symmetric, but not Archimedean, not radially symmetric, and without tail dependence. The theory is illustrated with numerical tables and graphics. A final part discusses the multi-dimensional variation of our original ratio form. The contributions are primarily theoretical, but they provide the framework for cutting-edge dependence models that have potential applications across a wide range of fields. Some established two-dimensional inequalities may be of interest beyond the purposes of this paper.

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

“…which is a special case of a ratio copula described in [37]. Now, let us remark that, for any (x, y) ∈ [0, 1] 2 , we have (1 − x a )(1 − y a ) ≥ 0, which implies that (x a + y a )/(1 + x a y a ) ≤ 1.…”

Extensions of Two Bivariate Strict Archimedean Copulas

“…We may mention the Gaussian copulas, Student copulas, extreme-value copulas, Farlie-Gumbel-Morgenstern (FGM) copulas, and Archimedean copulas. In addition, among the recent refrences, we may mention the new bivariate FGM copulas studied in [6] and [7], the bivariate copulas constructed via a unit Weibull distribution distortion in [8], the bivariate transformation of copulas proposed in [9], the bivariate copulas constructed via the Rüschendorf method in [10], the bivariate copulas based on the counter-monotonic shock method created in [11], the bivariate trigonometric-and hyperbolic-FGM-type copulas developed in [12], the bivariare exponential-type copulas developed in [13], the bivariate circular-linear copulas proposed in [14] and [15], and the bivariate ratio-type copulas elaborated in [16] and [17]. Recent applications of bivariate copulas in applied sectors include [18], [19] and [20].…”