A new rank correlation measure

A new rank correlation measure $\beta_n$ is proposed, so as to develop a nonparametric test of independence for two variables. $\beta_n$ is shown to be the symmetrized version of a measure earlier proposed by Borroni and Zenga (2007). More specifically, $\beta_n$ is built so that it can take the opposite sign, without changing its absolute value, when the ranking of one variable is reversed. Further, the meaning of the population equivalent of $\beta_n$ is discussed. It is pointed out that this latter association measure vanishes not only at independence but, more generally, at {\em indifference}, that is when the two variables do not show any ``tendency" to positive or negative dependence. The null distribution of $\beta_n$ needs an independent study: hence, the finite null variance and a table of critical values are determined. Moreover, the asymptotic null distribution of $\beta_n$ is derived. Finally, the performance of the test based on $\beta_n$ is evaluated by simulation. $\beta_n$ is shown to be a good competitor of some classical tests for the same problem.