Characterization of Differentiable Copulas

Abstract

This paper proposes a new class of copulas which characterize the set of all twice continuously differentiable copulas. We show that our proposed new class of copulas is a new generalized copula family that include not only asymmetric copulas but also all smooth copula families available in the current literature. Spearman's rho and Kendall's tau for our new Fourier copulas which are asymmetric are introduced. Furthermore, an approximation method is discussed in order to optimize Spearman's rho and the corresponding Kendall's tau.2010 Mathematics Subject Classification. 62H20.

Absolutely continuous copulas obtained by regularization of the Frechét--Hoeffding bounds

“…The question of regularity of copulas and the associated differentiability of copulas has been studied before and there are at least two well-known classes of differentiable copulas: Archimedean and Farlie-Gumbel-Morgenstern copulas. Recently a preprint claiming to characterize all twice differentiable copulas was presented [3]. In that paper also a new class, Fourier copulas, of twice differentiable copulas is introduced.…”