Bivariate copulas with cubic sections

Abstract

In this paper we present a method for construction families of bivariate copulas with cubic cross sections. We study dependence properties, measures of association, and concepts of symmetry for these copulas. Examples of both symmetric and asymmetric copulas with cubic sections are presented which extend some well known families of bivariate copulas (such as the iterated Farlie-Gumbel-Morgenstern, Kimeldorf and Sampson, Lin, and Sarmanov families of copulas) and which provide second-order approximations to the Frank and Plackett families of copulas.