Advances in multivariate Archimedean copula modeling

Copula functions are advanced and powerful tools that allow for the characterization of dependence structures between variables without being limited by their marginal distributions. The use of these functions started to gain popularity 30 years ago and has since become an applied tool in various fields. Several families of copulas exist, with the Archimedean family being one of the most useful due to its simple analytic form. A challenge concerning Archimedean copulas is that in most renowned members of the family, the dependence structure is defined by a single parameter that governs its properties, resulting in limited modeling flexibility. In this work, we present a method for enriching the dependence structure by adding parameters to the copula function. This is achieved by increasing the number of parameters of the inverse generator of the Archimedean copulas through compounding it with a density function of the dependence parameter. We prove that our new extended inverse generator generates a new Archimedean copula, and this method holds for any copula dimension. Furthermore, we demonstrate how our new method enhances the Clayton copula, one of the most renowned and widely used Archimedean copulas, by extending it from a single-parameter to a three-parameter function. This extension uses the generalized gamma distribution as the density function for the dependence parameter. As illustrated in the example, this specific enhancement significantly improves the fit of the copula to empirical data.