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== Kolmogorov–Smirnov statistic Kolmogorov-Smirnov统计==

== Kolmogorov–Smirnov statistic Kolmogorov-Smirnov统计==

where I_{[-\infty,x]}(X_i) is the indicator function, equal to 1 if X_i \le x and equal to 0 otherwise.

where I_{[-\infty,x]}(X_i) is the indicator function, equal to 1 if X_i \le x and equal to 0 otherwise.

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

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

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.

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.

 

F_{n}(x)={1 \over n}\sum _{i=1}^{n}I_{[-\infty ,x]}(X_{i})

where sup_''x'' is the [[supremum]] of the set of distances. By the [[Glivenko–Cantelli theorem]], if the sample comes from distribution ''F''(''x''), then ''D''_''n'' 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-Smirnov test#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.

K=\sup_{t\in[0,1]}|B(t)|

其中 {\displaystyle I_{[-\infty ,x]}(X_{i})}I_{[-\infty ,x]}(X_{i})是指标函数，如果 {\displaystyle X_{i}\leq x}X_{i}\leq x等于1，否则等于0。

K = sup _ { t in [0,1]} | b (t) |  

D_{n}=\sup _{x}|F_{n}(x)-F(x)|

==Kolmogorov distribution==

==Kolmogorov distribution==