On the epsilon-delta Structure Underlying Chatterjee's Rank Correlation

TL;DR

This paper offers an epsilon-delta interpretation of Chatterjee's rank correlation, linking it to local dependence and residuals, and introduces an L2 analogue that relates to Pearson's coefficient.

Contribution

It provides a novel epsilon-delta framework for understanding rank correlation measures and introduces an L2 version that connects to classical correlation coefficients.

Findings

Chatterjee's rank correlation is an empirical local L1 residual.

The epsilon-delta interpretation clarifies the dependence structure.

The L2 analogue recovers Pearson's coefficient under Gaussian assumptions.

Abstract

We provide an epsilon-delta interpretation of Chatterjee's rank correlation by tracing its origin to a notion of local dependence between random variables. Starting from a primitive epsilon-delta construction, we show that rank-based dependence measures arise naturally as epsilon to zero limits of local averaging procedures. Within this framework, Chatterjee's rank correlation admits a transparent interpretation as an empirical realization of a local L1 residual. We emphasize that the probability integral transform plays no structural role in the underlying epsilon-delta mechanism, and is introduced only as a normalization step that renders the final expression distribution-free. We further consider a moment-based analogue obtained by replacing the absolute deviation with a squared residual. This L2 formulation is independent of rank transformations and, under a Gaussian assumption, recovers Pearson's coefficient of determination.