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Assume $x_1, x_2, \dots, x_n$ belong to group $A$ and $y_1, y_2, \dots, y_m$ belong to group $B$.

We want to test if values in group $A$ are smaller than values in group $B$ as values in B.

I design the test statistics similar to Kendall's Tau by getting the pairs of $\mathrm{sign}(x_i-y_j)$ for $i= 1, 2, \dots, n$ and $j= 1, 2, \dots, m$.

As there are in total $n\times m$ pairs, let $$T= \frac{\sum_{i,j} \mathrm{sign}(x_i-y_j)}{m\times n},$$ where $T$ is between $[-1, 1]$.

My questions is that how do I get the test significance for $T?$

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  • $\begingroup$ If any of the $x_i$ equal or exceed any of the $y_j,$ wouldn't that contradict your hypothesis? What, then, would be the point of $T$? $\endgroup$
    – whuber
    Jun 18 '20 at 19:30
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For continuous rvs X and Y (i.e. absent ties), your test statistic is exactly equivalent to a Wilcoxon-Mann-Whitney test - indeed, it's linear in the usual U statistic.

So the easy way to test it is to use a Wilcoxon-Mann-Whitney with the same direction of null and alternative, the p-value should be the same.

Or you can just do a permutation test of your statistic.

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