Adequately modeling the dependence structure of high-dimensional random vectors is challenging. One typically faces a tradeoff between models that are rather simple but computationally efficient on the one hand, and very flexible dependence structures that become unhandy as the dimension of the problem increases on the other hand. Several popular families of copulas, especially when based on a factor-model construction, are extendible. Even though such structures are very convenient in large dimensions (due to the factor model / conditional i.i.d. structure), the assumption of conditional i.i.d. may be over-simplistic for real situations. One possibility to overcome extendibility without giving up the general structure is to consider hierarchical (or nested) extensions of the dependence structure in concern. Heuristically speaking, the dependence structure of hierarchical copulas is induced by some global stochastic factor affecting i.i.d. components and by additional group-specific factors that only affect certain sub-vectors. We present a survey of recent developments on hierarchical models, such as hierarchical Archimedean and Marshall-Olkin type dependence structures, and unify the literature by introducing the notion of h-extendibility. This definition generalizes extendible models in a natural way to hierarchical structures.      «

Adequately modeling the dependence structure of high-dimensional random vectors is challenging. One typically faces a tradeoff between models that are rather simple but computationally efficient on the one hand, and very flexible dependence structures that become unhandy as the dimension of the problem increases on the other hand. Several popular families of copulas, especially when based on a factor-model construction, are extendible. Even though such structures are very convenient in large dim...     »