Summary: K-S test has various limitations and Anderson-Darling or Cramer-von Mises would be better.
Notes:
[[Kolmogorov-Smirnov test]] is a nonparametric hypothesis test that measures the probability that a chosen univariate
Dataset is drawn from the same parent population as the second dataset (two-sample K-S test) or a continuous model (the one-sample K-S test)
K-S test is based on measuring the supremum (greatest) distance between the empirical distribution function (EDF) of a univariate dataset and the comparison step function of the second dataset (or ites cumulative distribution function)
The underlying distribution is assumed to be continuous.
Benefits of the K-S test is that it is distribution free, universally applied (no restrictions on sample size), critical values are widely available, can serve as goodness-of-fit test and easy to compute.
However, it is misleading in many ways as well.
K-S test require the EDFs to differ in a global fashion near the center of the distribution, but if the distributions are adjusted to have the same mean values
So if the EDFs cross each other multiple times and the maximum deviation is reduced, a better test would be the Cramer-von Mises (CvM) test
[[Cramer-von Mises test]] measures the sum of square deviations between the EDFs.
If the EDFs have differences near the beginning or end of distributions, K-S and CvM don't do well because the differences at the ends are small
[[Anderson-Darling test]] was developed as a weighted CvM test to overcome this
K-S test probabilities are wrong if the model was derived from the dataset
To overcome this, we can bootstrap the sample
The K-S test cannot be applied in 2 or more dimensions