Kendall Correlation Coefficient for non-Identically Distributed Variables

TL;DR

This paper introduces a theoretical Kendall correlation coefficient for non-identical bivariate data, proving its properties and convergence, supported by simulation experiments.

Contribution

It is the first to define and analyze a Kendall correlation coefficient for non-identically distributed variables, establishing its theoretical validity.

Findings

Expected value of rank Kendall correlation equals the theoretical coefficient.

The rank Kendall correlation converges in probability to the theoretical coefficient.

Simulation results support the theoretical findings.

Abstract

In the present paper, we discuss for the first time the theoretical Kendall correlation coefficient for non-identical bivariate data. In the non-identical case, we first introduce a theoretical Kendall correlation coefficient $\tau_n$ and show that the expected value of the rank Kendall correlation coefficient $\tilde{\tau}_n$ is equal to $\tau_n$. We then prove that $\tilde{\tau}_n$ converges in probability to $\tau=\lim_{n\rightarrow\infty} \tau_n$. These facts enable us to state that $\tau_n$ is a correctly defined theoretical Kendall correlation coefficient for the non-identical case. We also support our theoretical results by simulation experiments.