Singularity aspects of Archimedean copulas

Abstract

Calculating Markov kernels of two-dimensional Archimedean copulas allows for very simple and elegant alternative derivations of various important formulas including Kendall's distribution function and the measures of the level curves. More importantly, using Markov kernels we prove the existence of singular Archimedean copulas Aφ with full support of the following two types: (i) All conditional distribution functions y↦FxAφ(y) are discrete and strictly increasing; (ii) all conditional distribution functions y↦FxAφ(y) are continuous, strictly increasing and have derivative zero almost everywhere. The results show that despite of their simple analytic form Archimedean copulas can exhibit surprisingly singular behavior.