An ordering for the strength of functional dependence

Abstract: We introduce a new dependence order that satisfies eight natural axioms that we propose for a global dependence order. Its 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 dependence measures, such as Chatterjee's rank correlation, are increasing in this order. We characterize our ordering by the Schur order and by the concordance order, and we verify it in models such as the additive error model, the multivariate normal distribution, and various copula-based models.