Abstract: Chatterjee (2019) (see also Dette et al, 2013) introduced a novel rank based measure of dependence between X and Y real, which is 0 iff X and Y are independent, and 1 iff Y=h(X) for some h. Subsequent work by Cao and Bickel (2020) and Shi, Drton and Han (2020) pointed to poor local power properties for testing independence of this statistic as compared to classical procedures, such as those of Spearman, Blum, Kiefer and Rosenblatt and others. In many cases we show that statistics such as Chatteejee’s locally always lead to no power or misleading results. Measuring functional dependence is a different matter and we show, using some of Chatterjee’s ideas, how to construct simple rank based measures which have whatever local power properties we wish for independence testing and point to functional dependence with the same reliability as Chatterjee’s statistic.

Peter Bickel is a Professor Emeritus, and Professor of the Graduate School in the Department of Statistics at University of California, Berkeley. He is past President of the Bernoulli Society and of the Institute of Mathematical Statistics, a MacArthur Fellow, a COPSS prize winner, and a member of the American Academy of Arts and Sciences and of the National Academy of Sciences. He was awarded an honorary Doctorate degree from Hebrew University, Jerusalem in 1986, and from ETH, Zurich in 2014.

https://statistics.berkeley.edu/people/peter-bickel