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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?

kjetil b halvorsen
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Mathew Carroll
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  • 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? – Dave Jun 05 '21 at 16:29
  • The latter. Don't know what any of the $D_{i}$ are. – Mathew Carroll Jun 05 '21 at 16:30
  • Are the distributions totally independent of each other, or are you interested if the multivariate $x$ and $y$ are the same? – Dave Jun 05 '21 at 16:33
  • Ideally the latter. But a test that assumes independence would also be useful. – Mathew Carroll Jun 05 '21 at 16:34
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    "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. – Alexis Jun 05 '21 at 16:42
  • Thank you, I will consider the false discovery rate approach, but for this I assume $x_{1},...,$ and $x_{n}$ would need to be independent (similarly for the y's). Ideally, I wouldn't have to make this assumption. – Mathew Carroll Jun 05 '21 at 16:50
  • @MathewCarroll for the FDR they need not be independent, merely have [positive dependency](https://stats.stackexchange.com/questions/111756/the-meaning-of-positive-dependency-as-a-condition-to-use-the-usual-method-for). – Alexis Jun 05 '21 at 23:52
  • Interesting... it appears I will have some reading to do tomorrow. – Mathew Carroll Jun 06 '21 at 00:07

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