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The present contribution derives an explicit expression for (a version of)every uni- and multi-variate conditional distribution (i.e., Markov kernel) ofArchimedean copulas and uses this representation to generalize a recentlyestablished result, saying that in the class of multivariate Archimedeancopulas standard uniform convergence implies weak convergence of almost allunivariate Markov kernels, to arbitrary multivariate Markov kernels. Moreover,we prove that an Archimedean copula is singular if, and only if, almost alluni- and multivariate Markov kernels are singular. These results are thenapplied to conditional Archimedean copulas which are reintroduced largely froma Markov kernel perspective and it is shown that convergence, singularity andconditional increasingness carry over from Archimedean copulas to theirconditional copulas. As consequence the surprising fact is established thatestimating (the generator of) an Archimedean copula directly yields anestimator of (the generator of) its conditional copula. Building upon that, wesketch the use and estimation of a conditional version of a recently introduceddependence measure as alternative to well-known conditional versions ofassociation measures in order to study the dependence behaviour of Archimedeanmodels when fixing covariate values.

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