Characterization of Extreme Copulas

TL;DR

This paper characterizes the extreme points of the set of all n-dimensional copulas, showing they induce singular measures and providing constructions and conditions for extremality, with applications in various mathematical fields.

Contribution

It offers a new characterization of extreme copulas, including conditions and constructions, advancing understanding of their structure and applications in optimization and statistical theory.

Findings

Extreme copulas induce singular measures.

Constructed a subset of extreme copulas dense in the set.

Applications demonstrated in optimization and statistical modeling.

Abstract

In this paper our aim is to characterize the set of extreme points of the set of all n-dimensional copulas (n > 1). We have shown that a copula must induce a singular measure with respect to Lebesgue measure in order to be an extreme point in the set of n-dimensional copulas. We also have discovered some sufficient conditions for a copula to be an extreme copula. We have presented a construction of a small subset of n-dimensional extreme copulas such that any n-dimensional copula is a limit point of that subset with respect to weak convergence. The applications of such a theory are widespread, finding use in many facets of current mathematical research, such as distribution theory, survival analysis, reliability theory and optimization purposes. To illustrate the point further, examples of how such extremal representations can help in optimization have also been included.