Kolmogorov-Smirnov vs. Kuiper test


I would like to compare two 2D distributions with quantitative variables, illustrated here:

For each "x", they are several measures "y".
I can't assume these distributions are parametric.

A Kolgomorov-Smirnov test (p=0) and a Kuiper test (p=0.5) give rise to very different results.

Could you please tell me in which cases a Kolgomorov-Smirnov test is more appropriate than a Kuiper test, and vice versa?

Thank you very much


TS Contributor
That's hard to say. The two tests are essentially answering different questions. Which question is relevant to your analysis? Or, are both/neither relevant?
I'd like to test the null hypothesis according to which these two distributions are "the same".

I'd say both questions are relevant (or I wouldn't have reason for considering one of those more or less relevant than the other one).

What do you think? :)


TS Contributor
Then your two tests appear to say that there are differences in the central region, but not near the tail areas. This corresponds to what a visual examination of the data appears to show.
Thank you for your reply.

One clarification please:

Wikipedia indicates:
The trick with Kuiper's test is to use the quantity D+ + D− as the test statistic. This small change makes Kuiper's test as sensitive in the tails as at the median and also makes it invariant under cyclic transformations of the independent variable.
That seems to me different from what you wrote:
The K-S test is sensitive to shifts in the distribution (particularly around the median) while the Kuiper test is sensitive to changes in the tail areas.