On the Asymptotic Normality of Kendall's Rank Correlation Statistic

Abstract

Kendall's rank correlation statistic $T_n = \sum i > j \operatorname{sgn} (X_i - X_j)\cdot$ sgn $(Y_i - Y_j)$ is well known to be asymptotically normally distributed under the null hypothesis of independence as the sample size $n\rightarrow\infty$. In the note it is shown that this assertion can be obtained easily from the recurrence formula $p_n(t) = (1/n) \sum^n_{k =1}p_{n - 1}(t - 2k + n + 1)$ for the probability distribution $p_n$ of $T_n$ (see Kendall (1970), e.g.). This recurrence formula implies that $T_n$ has the same distribution as a sum of $(n - 1)$ well defined independent random variables to which the Lyapunov criterion applies.