New Axioms for Dependence Measurement and Powerful Tests

TL;DR

This paper introduces three new axioms for dependence measurement, improving theoretical foundations, and presents a new implementation for more powerful one-sided tests, with practical R code available.

Contribution

It proposes a new axiomatic framework for dependence measures, updates existing correlation concepts, and offers a novel testing method with practical implementation.

Findings

New axioms for dependence measurement based on three principles

Enhanced dependence measures like Vinod's R* demonstrate superior footing

A practical one-sided test using Taraldsen's exact density shows improved performance

Abstract

We build a context-free, comprehensive, flexible, and sound footing for measuring the dependence of two variables based on three new axioms, updating Renyi's (1959) seven postulates. We illustrate the superior footing of axioms by Vinod's (2014) asymmetric matrix of generalized correlation coefficients R*. We list five limitations explaining the poorer footing of axiom-failing Hellinger correlation proposed in 2022. We also describe a new implementation of a one-sided test with Taraldsen's (2021) exact density. This paper provides a new table for more powerful one-sided tests using the exact Taraldsen density and includes a published example where using Taraldsen's method makes a practical difference. The code to implement all our proposals is in R packages.