Efficiently sampling Archimedean copulas

Efficient sampling algorithms for both exchangeable and nested Archimedean copulas are presented. First, efficient sampling algorithms for the nested Archimedean families of Ali-Mikhail-Haq, Frank, and Joe are introduced. Second, a general strategy how to build a nested Archimedean copula from a given Archimedean generator is presented. Sampling this copula involves sampling an exponentiallytilted Stable distribution. For this task, a fast rejection algorithm is developed. It is proven for the more general class of tilted Archimedean generators that this algorithm reduces the complexity of the standard rejection algorithm to logarithmic complexity. As an application it is shown that this algorithm outperforms existing algorithms for sampling nested Clayton copulas. Third, with the additional help of randomization of generator parameters, explicit sampling algorithms for several nested Archimedean copulas based on different Archimedean families are found. As all algorithms work fast for large parameter ranges and do not require numerical inversion of Laplace transforms, they are recommendable for large-scale simulation studies, even in large dimensions. The presented ideas may also apply in the more general context of sampling distributions given by their Laplace-Stieltjes transforms.