# Two dimensional Kolmogorov-Smirnov test

For a one dimensional Kolmogorov-Smirnov test there is a theoretical result, one can compare his or her test results directly by theoretical one, which is infinite sum (notice this formula is for big samples http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test ), and this distribution is called a Kolmogorov distribution. And one can compute the significance level (p-value) directly from this CDF formula. Also it's possible to get a probability density distribution from this formular and many other things.

So my question : Is there a theoretical result for 2 dimensional distributions ? Or, if I compute a 2 dimensional kolmogorov-smirnov test myself, by code that I have written myself, how can I check the quality of my test result? In this case I would prefer to have theoretical information.

• There are some difficulties with a 2-D KS test. How exactly does your code work? (I don't want to see code, I want to understand what you told it to do - e.g. how do you compute the statistic?) – Glen_b May 17 '15 at 6:30
• As you know also this test has computational problems, but I've reduced the complexity of test, so I just compute on several points on the plane , and it's enough for Kolmogorov-Smirnov stastitics, but anyway there's nothing to compare with my results? At least any literature . !!!! Any table for big samples . !!! And also what do you mean by saying complexity, you mean there are no theory about 2 dimension?? I know that In cosmology , theoretical physicists use 2 dimensional kolmogorov-smirnov test , so how they compare their results ?? A lot of questions :) Thanks for response . – Narek May 18 '15 at 7:12
• Your first sentence is unclear. Yes, literature on multidimensional KS tests can readily be found. – Glen_b May 18 '15 at 10:34
• I dont think so . Can you say at least on source about two dimensional Kolmogorov-Smirnov test ? I mean theoretical results, not how to compute it . The first sentence ofcourse is unclear , because this is a long story to explain it properly . Thanks a lot . – Narek May 18 '15 at 10:44
• An example of a paper ... halweb.uc3m.es/esp/Personal/personas/dpena/articles/kolpl97.pdf – Glen_b May 18 '15 at 10:49