Bounds on Integrals with Respect to Multivariate Copulas

TL;DR

This paper extends the connection between copula bounds and the assignment problem from two to multiple dimensions, providing a generalized method and convergence results for integrals with respect to multivariate copulas.

Contribution

It generalizes the 2014 method linking copula bounds to the assignment problem from 2D to arbitrary dimensions, including convergence analysis.

Findings

Extended the copula-assignment connection to d-dimensions

Provided convergence statements for the generalized method

Applied the approach to three-dimensional dependence measures

Abstract

Finding upper and lower bounds to integrals with respect to copulas is a quite prominent problem in applied probability. In their 2014 paper, Hofer and Iaco showed how particular two dimensional copulas are related to optimal solutions of the two dimensional assignment problem. Using this, they managed to approximate integrals with respect to two dimensional copulas. In this paper, we will further illuminate this connection, extend it to d-dimensional copulas and therefore generalize the method from Hofer and Iaco to arbitrary dimensions. We also provide convergence statements. As an example, we consider three dimensional dependence measures.