Attia-3 A New Archimedean Bivariate Copula Based on Frailty Models for Modelling Positive Dependency Utilizing the Transformed Median Based Unit Rayleigh

In this paper, the author utilizes the frailty model to construct a parametric new Archimedean copula, Attia-3 copula. This copula depends on the transformed Median Based Unit Rayleigh (MBUR) distribution to an unbounded distribution defined on the interval from zero to infinity. In the paper, the joint PDF and CDF of the copula are derived for two bivariate distributions. The singularity of the copula is explained. The generator and inverse generator of the new copula are explored to depict the decreasing and convex nature of the generator. The Kendall tau measure of dependency is derived. Inference Function for Margin (IFM) is explored and utilized in real data analysis. The novelty of Attia-3 copula is that it is constructed using the frailty models that account for unobserved heterogeneity in the time to event data serving as a multiplicative element on the hazard rate. Attia-3 copula which is based on frailty model and cox proportional hazard model can only model positive dependency. The domain of the dependency parameter is (0,1]. The Kendall tau coefficient is , where is the dependency parameter. It can model upper and lower tail dependency, and . The upper and the lower tail coefficients are decreasing function in the parameter. The Kendall tau is also a decreasing function in the parameter. For a copula to model simultaneously upper and lower tail dependency with a single parameter without switching families or using mixture is very attractive for applications such as medical complications due to simultaneous deteriorations in organs, insurances losses as joint small-moderate losses, and environmental risks in the form of joint droughts are stronger than floods. The limitation of the Attia-3 is modelling only positive dependency and lack of singular part which denotes that the probability mass is concentrated along the diagonal signifying strong concurrences and permitting modelling ties that the continuous part fights to model. There is no product copula for this copula. Classifications: 62H20 This is a galley Proof under publication, waiting for Bank of America to transfer the APC to HDFC BANK LTD in India via SWIFT bank transfer for the sake of the Science Domain (Asian Research Journal of Mathematics)