Some Counterexamples Concerning Maximal Correlation and Linear Regression

TL;DR

This paper presents examples demonstrating that linear regression relationships between two variables do not guarantee that their maximal correlation equals their absolute correlation, challenging common assumptions in statistical dependence.

Contribution

It introduces specific counterexamples showing the disconnect between linear regression and maximal correlation, clarifying limitations of their relationship.

Findings

Maximal correlation can differ from absolute correlation despite linear regression.

Counterexamples challenge the assumption that linear regression implies maximal correlation equals absolute correlation.

Highlights limitations in using maximal correlation as a dependence measure in certain cases.

Abstract

A class of examples concerning the relationship of linear regression and maximal correlation is provided. More precisely, these examples show that if two random variables have (strictly) linear regression on each other, then their maximal correlation is not necessarily equal to their (absolute) correlation.