The Kolmogorov-Smirnov Test

One-Sample Test

This test is another goodness-of-fit test: Given a sample , we would like to test . By the Glivenko-Cantelli theorem, the empirical distribution function

converges uniformly to the actual distribution function of . So we choose

as our test statistic, which should be small (under ).

One can prove that for continuous , the null distribution of does not depend on , and can be calculated as follows:

where denote the order statistics.

One can even calculated the asymptotic distribution function for large : Let . Then, for ,