What non-parametric methods exist to assess whether two random samples belong to the same distribution?
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It depends, how do you want to define similarity of distributions? Same location, same spread (variance), same skewness/kurtosis, combinations of these, or just "differences"?
Non-parametric methods like the Mann-Whitney U test (Wilcoxon rank sum test) test the hypothesis that the two samples' distributions come from populations having the same location (the alternative hypothesis being that the two samples come from populations with different locations, e.g. one distribution has larger values than the other). In R, see
The Kolmogorov-Smirnov test considers differences in the distributions per se (so is sensitive to differences in location, variance, skewness, etc) of the two candidate distributions. In R, see