On differentiability and mass distributions of multivariate Archimedean copulas

Copulas, in particular Archimedean copulas are commonly viewed as analytically nice and regular objects. Motivated by a recently established result stating that the first partial derivatives of bivariate copulas can exhibit surprisingly pathological behavior, we focus on the class of d -dimensional Archimedean copulas denoted by C a r d and show that partial derivatives of order ( d − 1 ) can be surprisingly irregular as well. In fact, we prove the existence of Archimedean copulas C ∈ C a r d whose ( d − 1 ) -st order partial derivatives are pathological in the sense that for almost every x ∈ [ 0 , 1 ] d − 1 the derivative ∂ 1 . . . ∂ d − 1 C ( x , y ) does not exist on a dense set of y ∈ ( 0 , 1 ) . Since the existence of mixed partial derivatives of order ( d − 1 ) of a copula C is closely related to the existence of a discrete component of the corresponding Markov kernel K C , we also study mass distributions of Archimedean copulas. Building upon the interplay between Archimedean copulas and so-called Williamson measures we show how absolute continuity, discreteness and singularity of the Williamson measure propagates to the associated Archimedean copula. Moreover, considering the Lebesgue decomposition of the Markov kernel K C as well as the measures induced by these components and the marginal C 1 : d − 1 , we prove that the subclass of Archimedean copulas for which all three measures have full support, is dense in C a r d . Finally, viewing C a r d in the light of Baire categories, we show that, in contrast to the space of bivariate copulas, a topologically typical d -dimensional Archimedean copula C is not absolutely continuous but has a ( d − 1 ) -kernel K C whose discrete component is degenerate, implying that pathological elements are rare in C a r d .