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Is there any mathematical result that states that the Wilcoxon-Mann-Whitney (WMW) test is optimal in some sense, for a specific testing problem that is a subproblem of the general problem the WMW test is testing, say against an alternative of two specific distributions where one is stochastically larger than the other, maybe a location shift model with specified distributions but maybe something else? I have in mind maximum power for given level, however I'd be interested in other types of optimality as well. Also I suspect that any result would be asymptotic, maybe of the type "locally asymptotically optimal".

I had a look at the Hajek, Sidak, Sen book Theory of Rank Tests, but I don't think it has such a result. There is an exercise that states efficiency 1 of the Wilcoxon signed rank test for one sample in a specific situation, also mentioned here: https://www.jstor.org/stable/43686636

I am however not aware of anything like this for the two-sample test, and I'd like to know whether anything exists.

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The Wilcoxon test is locally asymptotically optimal for location shift in a logistic distribution. Because it's based on ranks, it is also locally asymptotically optimal whenever a monotone one:one transformation of the data would produce a location shift in a logistic distribution.

One place this is derived is example 13.14 of Asymptotic Statistics by van der Vaart.

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  • $\begingroup$ This is nice, thanks - I will let this open for longer just to see whether something else will come up. $\endgroup$
    – Christian Hennig
    Commented Apr 18, 2023 at 15:49
  • $\begingroup$ (It's also locally asymptotically optimal for data with a finite number of ordered categories under a proportional odds model, since it is exactly the score test for a binary predictor.) $\endgroup$
    – Thomas Lumley
    Commented Apr 18, 2023 at 22:49

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