Chatterjee (2021) introduced a simple new rank correlation coefficient that has attracted much recent attention. The coefficient has the unusual appeal that it not only estimates a population quantity first proposed by Dette et al. (2013) that is zero if and only if the underlying pair of random variables is independent, but also is asymptotically normal under independence. This paper compares Chatterjees new correlation coefficient to three established rank correlations that also facilitate consistent tests of independence, namely, Hoeffdings $D$, Blum-Kiefer-Rosenblatts $R$, and Bergsma-Dassios-Yanagimotos $tau^*$. We contrast their computational efficiency in light of recent advances, and investigate their power against local rotation and mixture alternatives. Our main results show that Chatterjees coefficient is unfortunately rate sub-optimal compared to $D$, $R$, and $tau^*$. The situation is more subtle for a related earlier estimator of Dette et al. (2013). These results favor $D$, $R$, and $tau^*$ over Chatterjees new correlation coefficient for the purpose of testing independence.
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