On Rank Correlation Coefficients

TL;DR

This paper introduces a new rank correlation coefficient $r_n$ that outperforms existing measures like Spearman's $ ho_{S,n}$ and Kendall's $ au_n$ in non-linear relationships, supported by theoretical and simulation analysis.

Contribution

A novel rank correlation coefficient $r_n$ is proposed, with properties analyzed and compared to existing coefficients, including theoretical variance and asymptotic performance.

Findings

$r_n$ performs better than other coefficients in non-linear cases.

Analytical expressions for $Var( au_n)$ and $Var(r_n)$ are derived.

Simulation results support the improved performance of $r_n$.

Abstract

In the present paper, we propose a new rank correlation coefficient $r_n$, which is a sample analogue of the theoretical correlation coefficient $r$, which, in turn, was proposed in the recent work of Stepanov (2025b). We discuss the properties of $r_n$ and compare $r_n$ with known rank Spearman $\rho_{S,n}$, Kendall $\tau_n$ and sample Pearson $\rho_n$ correlation coefficients. Simulation experiments show that when the relationship between $X$ and $Y$ is not close to linear, $r_n$ performs better than other correlation coefficients. We also find analytically the values of $Var(\tau_n)$ and $Var(r_n)$. This allows to estimate theoretically the asymptotic performance of $\tau_n$ and $r_n$.