On Kendall’s Correlation Coefficient

In this paper, the Kendall correlation coefficient is studied in the continuous case. In the beginning of the paper, we consider the Pearson correlation coefficient ρ and its sample analog ρn, which is a good approximation for ρ for large n since it converges to ρ in probability. We further discuss the Kendall rank correlation coefficient τn and its theoretical analog τ. In the continuous case, τn is defined in the terms of ranks of the concomitants of order statistics. It is shown that Eτn = τ and that τn converges in probability to τ. Thus, the coefficient τn is also a good approximation for τ, as is the case with the coefficients ρn and ρ. This detection explains the reason that τ can also be considered as a theoretical correlation coefficient. In many works, τ is used as a theoretical correlation coefficient without explanations as to why it can be considered as such. Since the coefficient τ has been little studied, we then discuss the basic properties of τ, its advantages and disadvantages and compare it with coefficient ρ. Among the advantages of τ we stress that τ exists for any continuous distributions. Some examples are given at the end of the paper.