Inference in multivariate Archimedean copula models

This paper proposes new rank-based estimators for multivariate Archimedean copulas. The approach stems from a recent representation of these copulas as the survival copulas of simplex distributions. The procedures are based on a reconstruction of the radial part of the simplex distribution from the Kendall distribution, which arises through the multivariate probability integral transformation of the data. In the bivariate case, the methodology is justified by the well known fact that an Archimedean copula is in one-to-one correspondence with its Kendall distribution. It is proved here that this property continues to hold in the trivariate case, and strong evidence is provided that it extends to any dimension. In addition, a criterion is derived for the convergence of sequences of multivariate Archimedean copulas. This result is then used to show consistency of the proposed estimators.