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Suppose $\mathbf{y}_{i} $ and $ \mathbf{x}_{i}\,$ are length-$m$ vectors and $D_{i}$ is some arbitrary distribution $(i=1,...,N)$. I would like to conduct the following hypothesis test:

$ H_{0}: \mathbf{x}_{1}, \mathbf{y}_{1} \sim D_{1}; \,\,\, \mathbf{x}_{2}, \mathbf{y}_{2} \sim D_{2}; ...;\,$ and $\, \mathbf{x}_{N}, \mathbf{y}_{N} \sim D_{N} $

$ H_{1}: H_{0}$ is false

I know $ H_{0}: \mathbf{x}, \mathbf{y} \sim D$ can be tested using the Kolmogorov-Smirnov test. But can this test be generalized as above?

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  • $\begingroup$ Do you know what all (or even any) of the $D_i$ are, or do you just want to know if any of the $N$ pairs differ? $\endgroup$
    – Dave
    Commented Jun 5, 2021 at 16:29
  • $\begingroup$ The latter. Don't know what any of the $D_{i}$ are. $\endgroup$
    – Mathew Carroll
    Commented Jun 5, 2021 at 16:30
  • $\begingroup$ Are the distributions totally independent of each other, or are you interested if the multivariate $x$ and $y$ are the same? $\endgroup$
    – Dave
    Commented Jun 5, 2021 at 16:33
  • $\begingroup$ Ideally the latter. But a test that assumes independence would also be useful. $\endgroup$
    – Mathew Carroll
    Commented Jun 5, 2021 at 16:34
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    $\begingroup$ "Multiple comparisons" are properly adjustments to $\alpha$ and rejection decision procedures. Sometimes they are expressed instead as transformations to $p$ values (i.e. $q$ values), but the latter has some incoherence, such as "probabilities" that do not lie between 0 and 1. Any collection of hypothesis tests based on a preferred family-wise error rate, or false discovery rate error may be adjusted for multiple comparisons. $\endgroup$
    – Alexis
    Commented Jun 5, 2021 at 16:42

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