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tests on ( rank ( X ) ,rank ( Y )) instead of on ( X,Y ), due to the distribution-free prop- erty of the tests on ( rank ( X ) ,rank ( Y )), comes at a cost of lower power, as noted by Sz´ ekely et al. (2007). Table 11 shows a power comparison of these two permuta- tion tests on ranks and on data. The results on ranks are not identical numerically (though very close) to those of Table 2 due to the use of a different seed to generate the data for these same settings. The power in most settings is indeed greater when the tests are used on data, and the maximal difference is almost 30% (in the Spiral setting for HHG) in favour of using the data. However, in some settings the power is actually larger for the test on ranks, e.g., in the Heavisine and 5Clouds settings for HHG, where the difference is 10% and 16%, respectively, in favour of using the ranked observations. Comparing the power of the HHG and dCov tests on data (in Table 11) to the power of our suggested minimum p -value statistic (in Table 2), we 

