Kendall Correlation Coefficient for non-Identically Distributed Variables

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.