A new family of copulas based on probability generating functions

Abstract

We propose a method to obtain a new class of copulas using a probability generating function (PGF) of positive-integer valued random variable. Some existing copulas in the literature are sub-families of the proposed copulas. Various dependence measures and invariant property of the tail dependence coefficient under PGF transformation are also discussed. We propose an algorithm for generating random numbers from the PGF copula. The bivariate concavity properties, such as Schur concavity and quasi-concavity, associated with the PGF copula are studied. Two new generalized FGM copulas are introduced using PGFs of geometric and discrete Mittag-Leffler distributions. The proposed two copulas improved the Spearman's rho of FGM copula by (-0.3333, 0.4751) and (-0.3333, 0.9573). Finally, we analyse a real dataset to illustrate the practical application of the proposed copulas. © 2024 Mathematical Institute Slovak Academy of Sciences.