On the failure of the bootstrap for Chatterjee's rank correlation

While researchers commonly use the bootstrap for statistical inference, many of us have realized that the standard bootstrap, in general, does not work for Chatterjee's rank correlation. In this paper, we provide proof of this issue under an additional independence assumption, and complement our theory with simulation evidence for general settings. Chatterjee's rank correlation thus falls into a category of statistics that are asymptotically normal but bootstrap inconsistent. Valid inferential methods in this case are Chatterjee's original proposal (for testing independence) and Lin and Han (2022)'s analytic asymptotic variance estimator (for more general purposes).