On the extensions of the Chatterjee-Spearman test

[Chatterjee (2021), ‘A New Coefficient of Correlation’, Journal of the American Statistical Association, 116(536), 2009–2022] introduced a novel independence test that is rank-based, asymptotically normal and consistent against all alternatives. One limitation of Chatterjee's test is its low statistical power for detecting monotonic relationships. To address this limitation, in our previous work (Zhang, 2024, Commun. Stat. – Theory Methods), we proposed to combine Chatterjee's and Spearman's correlations into a max-type statistic and established the asymptotic joint normality. This work examines three key extensions of the combined test. First, motivated by its original asymmetric form, we extend the Chatterjee-Spearman test to a symmetric version, and derive the asymptotic null distribution of the symmetrised statistic. Second, we investigate the relationships between Chatterjee's correlation and other popular rank correlations, including Kendall's tau and quadrant correlation. We demonstrate that, under independence, Chatterjee's correlation and any of these rank correlations are asymptotically joint normal and independent. Simulation studies demonstrate that the Chatterjee-Kendall test has better power than the Chatterjee-Spearman test. Finally, we explore two possible extensions to the multivariate case. These extensions expand the applicability of the rank-based combined tests to a broader range of scenarios.