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Suppose we have two populations X and Y, then compute the KS statistic/distance for samples drawn from them. Is this KS statistic/distance normally distributed? If so, how to prove it? If not, what distribution should it be? I know samples means $\bar{X}$ and $\bar{Y}$ are normally distributed according to the central limit theory.

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    $\begingroup$ you can't prove this because it's not true $\endgroup$
    – Aksakal
    Feb 6, 2022 at 18:05
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    $\begingroup$ The Kolmogorov distribution describes the relevant asymptotics when $X$ and $Y$ have identical continuous distributions. It clearly is non-Normal. $\endgroup$
    – whuber
    Feb 6, 2022 at 18:07
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    $\begingroup$ As a fairly general principle, distributions involving maxima won't be normal, nor even asymptotically normal. You typically need something more-or-less "mean-like". $\endgroup$
    – Glen_b
    Feb 6, 2022 at 23:02

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I'm assuming that you're talking about a case where X and Y are i.i.d. As noted, the distribution of the test statistic is not normal. Instead, it follows the Kologormov distribution. See the wikipedia entry: https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test#Kolmogorov_distribution

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there's no such result regarding K-S statistics applied to different populations. generally, it may not even be possible to calculate K-S stat. consider $X\sim\mathcal N$ and $Y\sim\chi^2$ - these two have different domains. how do you suppose to obtain K-S stat for them?

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    $\begingroup$ Not sure I understand the point you are making: ys both $\chi^2$ and $\mathcal{N}$ distributions have different supports (unless you just want to think of $P(\chi^2<t) = 0$ for all $t<0$, but the K-S statistic is built around the ordering of observations in each eCDF: in such a case as you describe the K-S statistic can be calculated, and the K-S null would almost certainly be correctly rejected, yes? Or am I misunderstanding? $\endgroup$
    – Alexis
    Feb 6, 2022 at 20:12
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    $\begingroup$ It’s possible and even expected to compare different domains in a KS test—when comparing two samples, their domains are discrete and different; when comparing a sample and a distribution, the sample’s domain is discrete and the distribution’s domain is usually continuous. $\endgroup$
    – Matt F.
    Feb 6, 2022 at 20:51
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    $\begingroup$ @MattF. I agree, but I think you are describing the one-sample K-S test (null: the sample is drawn from the specified i.i.d. null distribution), and the OP is asking about the two-sample K-S test (null: both samples are drawn from the same i.i.d. distribution, whatever that distribution is.). $\endgroup$
    – Alexis
    Feb 7, 2022 at 4:19

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