# Kolmogorov–Smirnov vs Brunner-Munzel

What are the main differences between the two-sample Kolmogorov–Smirnov test and the Brunner-Munzel test with respect to assumptions, power, etc? Both are non-parametric, and should work in the heteroscedastic case?

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I believe these tests are quite different. The Kolmogorov-Smirnov test is used to test if two samples come from the same distribution. The Brunner-Munzel test is a generalisation of the Behrens-Fisher problem with possible ties in the samples. The null hypothesis here is $H_0: p=P(X<Y)+0.5P(X=Y)=0.5$, where $X\sim F$ and $Y\sim G$. Yes, both should work in the heteroscedastic case. –  user10525 May 12 '12 at 18:23
@Procrastinator: Could you clarify what it means for the KS test to "work in the heteroscedastic case" given that this is a test on the equality of distributions from two iid samples? –  cardinal May 12 '12 at 20:17
@cardinal By 'both should work in the heteroscedastic case' I meant that the observations in each sample come from the same distribution. This is, you have two samples $x=(x_1,...,x_n)$ and $y=(y_1,...,y_m)$, where $x_j\sim F$ and $y_i \sim G$. Therefore there is no assumption about the variance (if it exists) of $x$ and $y$ (or the corresponding variables). Is this what you asked me to clarify? –  user10525 May 12 '12 at 20:37
@Procrastinator Thank you very much for your help! –  Alexander Serebrenik May 12 '12 at 20:59
Alexander: I think it would help to clarify what you mean by heteroscedastic. Normally I would think of that as variation within a set of observations, not differences in variation between two sets of observations (which, if I'm not mistaken, is how @Procrastinator appears to have interpreted it). Can you briefly clarify this? Saying the KS tests works in the former case does not make too much sense, but it can in the latter case (e.g., if you mean by "work" that it will reject the null hypothesis in large enough samples). Thanks. –  cardinal May 12 '12 at 21:02