A complete characterization of maximal copulas with a given track section

TL;DR

This paper introduces a new method to characterize maximal bivariate copulas with a given diagonal section, expanding the class of known solutions and addressing asymmetry more effectively than previous approaches.

Contribution

It proposes a novel order-theoretic approach to identify undominated copulas with prescribed diagonals, broadening the set of known solutions beyond existing bounds.

Findings

New copula construction method developed

Broader class of maximal copulas identified

Improved modeling of asymmetry in copulas

Abstract

Bivariate copulas with prescribed diagonal section were first studied by Bertino. Their maximality was studied so far only from the point of view of upper bounds which brings quasi-copulas into the picture and limits the resulting set substantially. We propose to study maximality of these families in the order theoretic sense. A copula C with given diagonal section {\delta} is called undominated if there is no copula C' {\neq} C with the same diagonal section {\delta} such that C {\leq} C'. The main contribution of this paper is a new method that provides copulas of the kind. This method generates a much wider class that contains the known upper bounds as a very small subclass. There was a recent call for the study of asymmetry which is addressed by our class better than by the known ones. Corresponding quasi-copulas can be obtained from our copulas via splicing techniques. Most results are given on the level of tracks.