Archimedean Copulas

In this chapter, we explore the world of bivariate Archimedean copulas (AC). We start with Sect. 2.1 by reviewing the copula families discussed in the previous chapter, which were constructed using Sklar’s theorem, probabilistic reasoning, and mixture copulas. However, the members of the Archimedean family are not constructed using Sklar’s theorem directly. Instead, they are related to Laplace–Stieltjes transforms (LSTs). We introduce the concepts of LSTs and completely monotone functions, which play a crucial role in defining Archimedean copulas. Additionally, we address the concept of d-monotonicity as a weaker version of complete monotonicity, which allows for more flexibility in certain cases. In Sect. 2.2, we delve into the Archimedean property and its connection to triangular norms. We discuss how bivariate Archimedean copulas are linked to a special class of triangular norms used to represent logical conjunctions. Furthermore, in Sect. 2.3, we describe parametric families of Archimedean copulas. We refer to the works of Nelsen (2006) and Joe (2014) for a comprehensive list of popular generators and their properties. We highlight the Gumbel, Clayton, Frank, and Joe copulas, discussing their respective generator functions, inverse functions, copula formulas, and their relations to Kendall’s \(\tau \) . We also provide visualizations of the density functions of these copulas for various marginal distributions. Lastly, we introduce the concept of outer power transformation, which allows for the modification of existing copula families. By fixing a copula family and applying the outer power transformation, we can adjust the properties of the resulting copulas. This provides additional flexibility, which is particularly useful for tail dependence modeling.