Following a comment from efrique pointing out that this statistic is called Spearman footrule, I want to clarify the notation in
namely (a) that the ranks of and are considered for the whole sample, i.e.
instead of being computed separately for the ‘s and the ‘s, and then (b) that the ranks are reordered for each group (meaning that the groups could be of different sizes). This statistics is therefore different from the Spearman footrule studied by Persi Diaconis and R. Graham in a 1977 JRSS paper,
where and are permutations from . The mean of is approximately . I mistakenly referred to Spearman’s rank correlation test in the previous post. It is actually much more related to the Siegel-Tukey test, even though I think there exists a non-parametric test of iid-ness for paired observations… The ‘s and the ‘s are thus not paired, despite what I wrote previously. This distance must be related to some non-parametric test for checking the equality of location parameters.