An ordering for the strength of functional dependence

TL;DR

This paper introduces the conditional convex order, a new dependence order that unifies and characterizes various dependence measures and properties, providing a comprehensive framework for understanding functional dependence.

Contribution

It defines the conditional convex order, links it to existing measures, and demonstrates its properties across multiple statistical models, offering a unified perspective.

Findings

The order characterizes independence and perfect dependence.

Various dependence measures are increasing in this order.

The order applies to models like Gaussian and copula-based models.

Abstract

We introduce a new dependence order, termed the conditional convex order, whose minimal and maximal elements characterize independence and perfect dependence. Moreover, it characterizes conditional independence, satisfies information monotonicity, and exhibits several invariance properties. Consequently, it is an ordering for the strength of functional dependence of a random variable Y on a random vector X. As we show, various recently studied dependence measures -- including Chatterjee's rank correlation, Wasserstein correlations, and rearranged dependence measures -- are increasing in this order and inherit their fundamental properties from it. We characterize the conditional convex order by the Schur order and by the concordance order, and we verify it in settings such as additive error models, the multivariate normal distribution, and various copula-based models. Our results offer a unified perspective on the behavior of dependence measures across statistical models.