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 $\tau ={\lim }_{n\to \mathrm{\infty }}{\tau }_n$. 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.