Geometry of Discrete Copulas

Multivariate distributions are fundamental to modeling. Discrete copulas can be used to construct diverse multivariate joint distributions over random variables from estimated univariate marginals. The space of discrete copulas admits a representation as a convex polytope which can be exploited in entropy-copula methods relevant to hydrology and climatology. To allow for an extensive use of such methods in a wide range of applied fields, it is important to have a geometric representation of discrete copulas with desirable stochastic properties. In this paper, we show that the families of ultramodular discrete copulas and their generalization to convex discrete quasi-copulas admit representations as polytopes. We draw connections to the prominent Birkhoff polytope, alternating sign matrix polytope, and their most extensive generalizations in the discrete geometry literature. In doing so, we generalize some well-known results on these polytopes from both the statistics literature and the discrete geometry literature.