Kolmogorov–Smirnov test

The Kolmogorov-Smirnov test is used to determine whether two empirical distributions are different or whether an empirical distribution differs from a theoretical distribution.

The empirical cumulative distribution for N observations y_i is defined as E(x) = Σ _i (y_i < x). The two one-sided Kolmogorov-Smirnov test statistics statistics are given by

D_N⁺ = MAX( E(x) - F(x) )

D_N^- = MAX( F(x) - E(x) )

where F(x) is the theoretical distribution or another empirical distribution. Knuth gives a detailed description of how to analyze the significance of this pair of statistics. Many people use MAX(D_N⁺, D_N^-) instead, but the distribution of this statistic is more difficult to deal with.

Note that when the underlying independent variable is cyclic as with day of the year or day of the week, then Kuiper's test is more appropriate. Numerical Recipes is again a good source of information on this. Note: Kuiper is pronounced in Dutch the way an English persion would pronounce Cowper.

Links:

See http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm for a lovely explanation of the one-sided KS test and http://www.io.com/~ritter/JAVASCRP/NORMCHIK.HTM for Javascript code that implements both the one-sided and two-sided tests. As always, Numerical Recipes ISBN: 0521431085 is a prime resource for this sort of thing (see http://www.nr.com/nronline_switcher.html for a discussion.