On convergence and singularity of conditional copulas of multivariate Archimedean copulas, and conditional dependence

Abstract

The present paper derives an explicit expression for (a version of) every uni- and multivariate conditional distribution (i.e., Markov kernel) of Archimedean copulas and uses this representation to generalize a recently established result, saying that in the class of multivariate Archimedean copulas standard uniform convergence implies weak convergence of almost all univariate Markov kernels, to arbitrary multivariate Markov kernels. Moreover, it is proved that an Archimedean copula is singular if, and only if, almost all uni- and multivariate Markov kernels are singular. These results are then applied to conditional Archimedean copulas which are reintroduced largely from a Markov kernel perspective and it is shown that convergence, singularity and conditional increasingness carry over from Archimedean copulas to their conditional copulas. As a consequence, the surprising fact is established that estimating (the generator of) an Archimedean copula directly yields an estimator of (the generator of) its conditional copula. Building upon that, we sketch the use and estimation of a conditional version of a recently introduced dependence measure as alternative to well-known conditional versions of association measures in order to study the dependence behavior of Archimedean models when fixing covariate values.