On Kendall’s correlation coefficient

Abstract

In the present paper, the Kendall correlation coefficient is studied in the continuous case. In the beginning of the paper, we discuss the Pearson correlation coefficient $\rho$ and its statistical analogue $\rho_n$, which is a good approximation for $\rho$ for large n since it converges to $\rho$ in probability.We further discuss the Kendall rank correlation coefficient $\tau_n$ and its theoretical analogue $\tau$ . In the continuous case, $\tau_n$ is defined in the terms of ranks of concomitants of order statistics. It is shown that $E\tau_n$ = $\tau$ and that $\tau_n$ converges in probability to $\tau$. That way, $\tau_n$ is also a good approximation for $\tau$, as well as it is for the coefficients $\rho_n$ and $\rho$. This finding explains why $\tau$ can also be considered a theoretical correlation coefficient. In many works $\tau$ was already used as a theoretical correlation coefficient without explanation why it can be considered as such. Since coefficient $\tau$ has been little studied, we then discuss the basic properties of $\tau$ , advantages and disadvantages, compare it with coefficient $\rho$. Amongst the advantages of $\tau$ we note that $\tau$ exists for any continuous distributions. At the end of this work, we present some illustrative examples.

Keywords:

bivariate distributions, concomitants of order statistics, Pearson’s and Kendall’s, correlation coefficients