I have been given a homework in a subject called "Non-Parametric Statistics" and I'm a bit stuck with it. I would be very thankful if you could give me any advice or help, which would lead to a solution!

The task is about Wilcoxon Rank Sum Test, but in a case, where there are equal observations in the sample. In this case midranks are being used (e.g. if sample consists of 1, 3, 3, 5. Then the ranks would be 1, (2+3)/2 = 2,5, (2+3)/2 = 2,5, 4. As we see, the distribution of $W_s^{*}$ is symmetrical, because mean, mode and median all fall at the same point (which is 3 at the moment). And the formula for a Wilcoxon Rank Sum Test in this case is: $$ W_{s}^{*} = \sum_{i=1}^{n} midrank(X_i) $$ ($X$ and $Y$ are observations from a united variations series.)

The task is as follows:

Bring an example, where the distribution of $W_{s}^{*}$ is NOT symmetrical under H0 hypothesis. (An example must have realistic subject - what is being examined and measured etc!)

  • $\begingroup$ What rule do you use to rank tied values? $\endgroup$ – whuber Nov 8 '18 at 15:56
  • $\begingroup$ I'm not sure how to explain it for general case, so I'll do it by example. Hope you understand it. If we have a variation series of 1; 3; 3; 5. Then the ranks are 1 (obvious), (2+3)/2 = 2.5, (2+3)/2 = 2.5, 4. So for tied values we add the places on which we have the tied values (at the moment on the 2nd and the 3rd place) and then we divide it by the count of tied values. $\endgroup$ – Martin Smith Nov 8 '18 at 16:07
  • $\begingroup$ Thank you. That ought to suggest an answer to the question. $\endgroup$ – whuber Nov 8 '18 at 16:15
  • $\begingroup$ Sorry, but in which sense? $\endgroup$ – Martin Smith Nov 8 '18 at 19:57

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