Exploring Two Modified Ali-Mikhail-Haq Copulas and New Bivariate Logistic Distributions

Abstract

The Ali-Mikhail-Haq copula is a bivariate ratio-type Archimedean copula known for its simplicity and flexibility in modeling moderate negative and positive dependence structures, making it widely used in various fields. However, it is limited to capturing asymmetric dependence features, significant negative correlations, or versatile tail dependence properties. This research paper proposes two modifications of the Ali-Mikhail-Haq copula that overcome these limitations, but at the price of the loss of the positive dependence nature. Contrary to the common approach that focuses on modifying the corresponding generator function, we apply direct functional changes to the Ali-Mikhail-Haq copula. We thus perturb its Archimedean identity. The first copula has the particularity of being non-exchangeable, capable of reaching an interesting level of negative dependence correlations, and possessing flexible tail dependence properties. The second copula offers another modeling option; it is exchangeable like the Ali-Mikhail-Haq copula, but it benefits from a broader range of negative dependence correlations and more adaptable tail dependence properties. For the two proposed copulas, we investigate their main characteristics, including quadrant dependence, copula density function, conditional copulas, couples of value generation, extended variants via standard copula schemes, comprehensive copula orders, and weighted harmonic mean copula transformations. An application on two new bivariate logistic distributions in a two-component system context is given. When possible, numerical and graphical studies are given to strengthen the theory.

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.…”