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Since it is not easy to apply a Multivariate Kolmogorov-Smirnov test for two samples, I thought: is it possible to compare the different statistics with the univariate KS? For example:

Object 1: statistic 1, statistic 2, etc.

Object 2: statistic 1, statistic 2, etc.

If I want to check if these 2 objects are the same, can I just use the univariate KS test for two samples and compare first statistics 1, then 2, and so on? Then if all tests say that all the statistics are the same, does it mean that objects are the same?

Of course if the statistics have the same "number" it means that they measure the same thing.

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What you're doing is comparing marginal distributions.

Equality of marginal distributions doesn't imply equality of joint distribution.

Specifically, the copula can be different in each of the two cases, while the margins remain the same.

Some examples (for normal margins) are described here, but the issue applies much more generally.

Even having bivariate margins equal doesn't make a trivariate distribution the same. See here for an example.

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  • $\begingroup$ Ok, thank you, I think I see the problem. Do you know which kind of two sample test for multivariate data could I apply? Suppose I know nothing beforehand about the distributions. $\endgroup$ Commented Sep 21, 2014 at 10:58
  • $\begingroup$ Added a couple of links you might find helpful. $\endgroup$
    – Glen_b
    Commented Sep 21, 2014 at 11:00
  • $\begingroup$ Do you know about a test for non normal margins (non parametric test)? $\endgroup$ Commented Sep 25, 2014 at 9:52
  • $\begingroup$ Sure, any univariate goodness of fit test for normality. A common test for normality vs non-normality is the Shapiro-Wilk - but that invites the important question ... why test that? $\endgroup$
    – Glen_b
    Commented Sep 25, 2014 at 9:55
  • $\begingroup$ I meant about a multivariate non parametric test for comparing two samples and test if they are the same or not. $\endgroup$ Commented Sep 25, 2014 at 9:57

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