Exploring a new two-parameter Archimedean copula: the Gumbel-Joe copula

In this article, we present a novel Archimedean copula constructed from a unique strict generator function. It can be described as a two-parameter unification of the well-established Gumbel-Barnett and Joe copulas. The first part is devoted to its formulation, as well as those of the corresponding density, the conditional copula, and the Kendall distribution function. Graphs are also included to illustrate their shape behavior under different parameter configurations. In a second part, we examine some of its notable properties, with emphasis on the correlation properties. Practical applications are discussed in the final part. In particular, we use the maximum likelihood estimation method to determine the unknown parameters involved from the data and perform a simulation study to demonstrate the effectiveness of this approach. We also analyze a dataset to provide practical illustrations of copula behavior and potential.