Bayesian nonparametric mixtures of asymmetric Archimedean copulas

Abstract

Archimedean copulas are exchangeable: a single generator parameter controls every pairwise association, and the copula is invariant under permutations of its arguments. Even when mixtures of Archimedean copulas are formulated, the exchangeability feature is inherited. Non exchangeable copulas can be defined through the Khoudraji transformation of an Archimedean copula. This relies in replacing the arguments by a power transformation and adding a factor based on the independence copula. The construction produces pairwise Kendall tau coefficients that depend on the power parameters and recovers the original Archimedean copula as a particular case. We propose a Bayesian nonparametric mixture model by mixing over the copula generator parameter and the parameters that characterize the asymmetry. We use the Poisson--Dirichlet process as a mixing distribution, which groups observations by both strength and direction of dependence. We derive the copula density, the pairwise association measures, the tail coefficients, and the full posterior conditional distributions to perform a Bayesian analysis. We carry out simulation studies and real data analyses. The results show that the Khoudraji mixture improves greatly on, or agrees with, the exchangeable fit.