Properties of HierarchicalArchimedean Copulas

In this paper we analyse the properties of hierarchical Archimedean copulas. This classis a generalisation of the Archimedean copulas and allows for general non-exchangeable dependencystructures. We show that the structure of the copula can be uniquely recovered from all bivariate margins.We derive the distribution of the copula value, which is particularly useful for tests and constructingcon¯dence intervals. Furthermore, we analyse dependence orderings, multivariate dependence measuresand extreme value copulas. Special attention we pay to the tail dependencies and derive several taildependence indices for general hierarchical Archimedean copulas.