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Statistical methods like wilcox.test to t.test test differences in medians or means. I would like to compare the difference in extreme percentile (could be one side or both size). I don't know how to define extreme percentile in the best way, as it could be just 1% but it may miss any changes in 5%. So the definition of extreme percentile has to be factored in the solution to the problem.

The distributions in the regions around the middle are not necessarily exactly same but should close. Even they are a little different, I don't really care as I just want to tell whether there is any significant difference be in the tail regions.

Could anybody suggest methods for doing so? Thanks.

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    $\begingroup$ You should start by defining what a "tail" might be. Would you be referring to an extreme percentile? To a mean plus a given multiple of the standard deviation? What really is the problem you are trying to solve? $\endgroup$
    – whuber
    Apr 3, 2018 at 21:44
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    $\begingroup$ You should also mention if the distributions can be assumed to have the same shape. $\endgroup$ Apr 4, 2018 at 0:03
  • $\begingroup$ @whuber let me know if my changes are clear. $\endgroup$ Apr 4, 2018 at 16:11
  • $\begingroup$ Wilcox-Mann-Whitney (wilcox.test) is not a test of medians, except in certain cases, and so it may respond to information in the tails. ... If you want to test for differences in a specific quantile, there are a few approaches I think might work. Have you looked at quantile regression? I'm not sure how well that works with extreme quantiles though. A couple other ideas: Mood's median test with the median replaced with the quantile of interest; and a permutation test comparing the two sets for the quantile of interest. $\endgroup$ Apr 5, 2018 at 20:03

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