Attia-1 and Attia-2 New Archimedean Bivariate Copulas Modeling Positive Dependency

In this paper, the author introduces new methods to construct Archimedean copulas. The generator of each copula fulfills the sufficient conditions as regards the boundary and being continuous, decreasing, and convex. Each inverse generator also fulfills the necessary conditions as regards the boundary conditions, marginal uniformity, and 2-increasing properties. Although these copulas satisfy these conditions, they have some limitations. They do not cover the entire dependency spectrum, ranging from perfect negative dependency to perfect positive dependency, passing through the state of independence. Both copulas exhibit positive dependency and upper tail dependency, but neither has lower tail dependency. The product copula is present for each of them. For each copula, the author discusses the derivation, the properties, whether it has a singular part or not, the Kendall tau measure for dependency, and the upper and lower tail dependency. The article shows figures for depicting the joint CDF, and joint PDF for each copula. For each copula, inference is supported by real data analysis using the inference function for margins (IFM) for estimation. The new copulas, Attia-1 and Attia-2 inherent both the limitations and the advantages of the classical Archimedean copulas as discussed in the introduction. The two new copulas and the Gumbel copula exhibit similarity for modelling weak positive association and upper tail dependence. However, both new copulas can add new substantial contributions to the field of copula theory. The new copula, Attia-1 has a domain (0,1], while Attia-2 copula has a domain [-1,1]-{0}. The Attia-1 copula allows the dependency parameter exploration near the zero adding finer control of the weak positive association and extreme upper tail co-occurrences. Attia-2 copula allows the dependency parameter exploration near the zero and negative dependency parameter adding finer control over the weak positive association and the extreme upper tail co-occurrence. The Gumbel copula also model the weak positive association and upper tail dependency but has a parameter domain that is bounded below by 1. So the new copulas extend the domain of modelling weak positive association and upper tail dependency that are encountered in many medical and financial datasets where the variables are mostly independent but extreme values can co-occur. This enhances the flexibility of Archimedean copula. The Attia-2 copula can be generalized by replacing the (2) in the exponent by a free parameter to partially decouple the global dependency from the tail dependency.  This also adds more flexibility to the Archimedean copula.