Quantitative analysis of non-exchangeability in bivariate copulas: Sharp bounds, statistical tests and mixing constructions

TL;DR

This paper quantifies non-exchangeability in bivariate copulas, establishing bounds, constructing copulas with prescribed asymmetry, and proposing a permutation test for exchangeability with finite-sample guarantees.

Contribution

It introduces a framework for measuring and constructing non-exchangeable copulas, linking asymmetry to dependence measures, and develops a nonparametric test for exchangeability.

Findings

Sharp bounds relate non-exchangeability to dependence measures.

Explicit construction of copulas with any prescribed asymmetry.

A permutation test with exact finite-sample error control.

Abstract

This paper studies the degree to which a bivariate copula fails to be symmetric under coordinate permutation, a property known as non-exchangeability. Working within an axiomatic framework that quantifies this asymmetry through a family of $L^p$-based measures, we establish sharp bounds linking non-exchangeability to classical dependence and concordance measures, prove exact scaling laws under convex mixing that enable explicit construction of copulas with any prescribed degree of asymmetry, and characterise the class of maximally non-exchangeable copulas together with the feasible range of asymmetry--concordance pairs. On the inferential side, we propose a nonparametric permutation test for exchangeability with exact finite-sample error control and consistency against all asymmetric alternatives, validated by Monte Carlo simulation and illustrated on a real data set.