Editing Kolmogorov–Smirnov test (section)

==One-sample Kolmogorov–Smirnov statistic==
The [[empirical distribution function]] ''F''<sub>''n''</sub> for ''n'' [[Independent and identically distributed random variables|independent and identically distributed]] (i.i.d.) ordered observations ''X<sub>i</sub>'' is defined as

<math display="block">
F_{n}(x)=\frac{\text {number of (elements in the sample} \leq x)}{n}=\frac{1}{n} \sum_{i=1}^{n} 1_{(-\infty,x]}(X_{i}),
</math>
where <math>1_{(-\infty,x]}(X_i)</math> is the [[indicator function]], equal to 1 if <math>X_i \leq x</math> and equal to 0 otherwise.

The Kolmogorov–Smirnov [[statistic]] for a given [[cumulative distribution function]] ''F''(''x'') is

<math display="block">D_n= \sup_x |F_n(x)-F(x)|</math>

where sup<sub>''x''</sub> is the [[supremum]] of the set of distances. Intuitively, the statistic takes the largest absolute difference between the two distribution functions across all ''x'' values.

By the [[Glivenko–Cantelli theorem]], if the sample comes from the distribution ''F''(''x''), then ''D''<sub>''n''</sub> converges to 0 [[almost surely]] in the limit when <math>n</math> goes to infinity. Kolmogorov strengthened this result, by effectively providing the rate of this convergence (see [[#Kolmogorov distribution|Kolmogorov distribution]]). [[Donsker's theorem]] provides a yet stronger result.

In practice, the statistic requires a relatively large number of data points (in comparison to other goodness of fit criteria such as the [[Anderson–Darling test]] statistic) to properly reject the null hypothesis.