Characterization of Differentiable Copulas

TL;DR

This paper introduces a new generalized class of twice continuously differentiable copulas, encompassing existing smooth and asymmetric copulas, and discusses methods to optimize dependence measures like Spearman's rho and Kendall's tau.

Contribution

It proposes a novel class of copulas that unify and extend existing smooth copula families, including asymmetric ones, with methods to optimize dependence measures.

Findings

The new copula class includes all smooth copulas in literature.

Spearman's rho and Kendall's tau are derived for the new Fourier copulas.

An approximation method for optimizing dependence measures is discussed.

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.