Extreme Points of Polytopes of Discrete Copulas

Abstract

Discrete copulas are useful tools in statistics to represent the joint distribution of discrete random vectors. Furthermore, they are fascinating mathematical objects that admit a representation as a convex polytope. In this work, we analyze the set of extreme points of convex polytopes of discrete copulas. We focus on the general class of discrete copulas defined on arbitrary grid domains of the unit square, thereby providing a complete characterization of extremal discrete copulas in the most general setting. To do so, we exploit well-known results on the extreme points of transportation polytopes. Finally, we show that the characterization presented here generalizes previous work on the topic.