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  • $\begingroup$ This answer is, unfortunately, incorrect in several ways and also mildly incoherent. First, in your opening sentence, you have "...how you expect that Wilcoxon Mann Whitney U (MW-U) would be more robust?", whereas in the second sentence of part 2, you state "...MW-U test, in its fundamental form, ... is, by definition, robust." Hmmm... However, the second statement is false; distribution-free in no way implies robustness. Consider the sample mean as a counterexample to this. As Peter Huber observed, robust, distribution-free, and nonparametric are actually not closely related properties. $\endgroup$
    – jbowman
    Commented Apr 29 at 21:33
  • $\begingroup$ You then contradict yourself later in the paragraph when you state that "...the Student t test suffers from $\alpha$-error inflation when the variances are different ... and so does the MW-U", which implies the test is not in fact robust against differences in variances. Furthermore, your initial statement in part 1 ignores the fact that the two null hypotheses are equivalent when there is only a shift in location between the two distributions, or, in the case of the Exponential, a shift in scale. In these cases, and others that can easily be constructed, the nulls are equivalent. $\endgroup$
    – jbowman
    Commented Apr 29 at 21:40
  • $\begingroup$ Can you provide your definition of "robustness"? How do you evaluate it? Then maybe we can discuss whether a test is, or not, robust. My statements stem from the fact that the OP seems to be comparing the power of the 2 tests (since he creates samples which are not compatible with the Null), and not the robustness. And a test which makes no distributional assumptions is by definition, robust. It may have other flaws etc., but it does not care about breaking the assumptions because it makes none... If a test is robuts by definition, how can one test that robustness; simple logic. $\endgroup$
    – jginestet
    Commented Apr 29 at 21:47
  • 1
    $\begingroup$ Power is indeed part of it, with the level being the other part. If we didn't care about power when the assumptions of a test are violated, we'd save ourselves the effort and just generate a $U(0,1)$ variate, rejecting when $u \leq 0.05$ or some such, because the assumptions are (almost) always violated to some degree. Studying what happens to the power of a test as assumptions are violated has a long history; I find it in Robust Statistics (Hampel et. al.), my 1986 edition, and I'm quite sure it wasn't novel then. $\endgroup$
    – jbowman
    Commented Apr 29 at 22:54
  • 1
    $\begingroup$ 1. Someone learning statistics from Wikipedia has much more learning to do. 2. You are changing the subject to one of my definition of robustness, and throughout this thread, and the other thread we are engaged on, you fail to address any of the points I make. 3. A trivial example of applying the Wikipedia definition to a distribution-free test is a violation of the i.i.d. assumption; consider the usual distribution-free test of the location of the median based on the order statistics. Its performance breaks down as the correlation between observations increases. $\endgroup$
    – jbowman
    Commented May 2 at 18:58