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We are given n classes of real values. The number of elements in each class is not too high and can be different between classes.

We want to test whether there exist a class with significantly higher (average) values from all other classes or not. The case is somehow similar to ANOVA, but slightly different.

What test do you propose?

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It seems you're interested in something along the lines of:

$H_0: \mu_1=\mu_2=...=\mu_k$
vs.
$H_1: \mu_j>\mu_1,...,\mu_{j-1},\mu_{j+1},...,\mu_k\quad$ for some $j$

  1. There's no great difficulty in constructing a permutation test for this; for example, consider the statistic "largest sample mean minus the mean of the remaining samples", and then performing a one-tailed test.

  2. An alternative would be to frame it as a set of one-sided ANOVA contrasts (not orthogonal) of the form $c_i'\mu$ (e.g. $c_3'\,\propto\,(-1, -1, 9, -1, -1, -1, -1, -1, -1, -1)\,$) and test them all one-tailed, presumably with some adjustment for multiple testing.

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