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The Kolmogorov distribution is the distribution of the [[random variable]]

The Kolmogorov distribution is the distribution of the [[random variable]]

where B(t) is the Brownian bridge. The cumulative distribution function of K is given by

where B(t) is the Brownian bridge. The cumulative distribution function of K is given by

\operatorname{Pr}(K\leq x)=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)},

\operatorname{Pr}(K\leq x)=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)},

which can also be expressed by the Jacobi theta function \vartheta_{01}(z=0;\tau=2ix^2/\pi). Both the form of the Kolmogorov–Smirnov test statistic and its asymptotic distribution under the null hypothesis were published by Andrey Kolmogorov, while a table of the distribution was published by Nikolai Smirnov. Recurrence relations for the distribution of the test statistic in finite samples are available.

\operatorname{Pr}(K\leq x)=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)},

 

\operatorname{Pr}(K\leq K_\alpha)=1-\alpha.\,

\operatorname{Pr}(K\leq K_\alpha)=1-\alpha.\,

[[convergence of random variables|in distribution]], where ''B''(''t'') is the [[Brownian bridge]].

[[convergence of random variables|in distribution]], where ''B''(''t'') is the [[Brownian bridge]].

在其分布中，B（t）指的是布朗桥。

在其分布中，B（t）指的是布朗桥。

If ''F'' is continuous then under the null hypothesis <math>\sqrt{n}D_n</math> converges to the Kolmogorov distribution, which does not depend on ''F''. This result may also be known as the Kolmogorov theorem. The accuracy of this limit as an approximation to the exact cdf of <math>K</math> when <math>n</math> is finite is not very impressive: even when <math>n=1000</math>, the corresponding maximum error is about <math>0.9\%</math>; this error increases to <math>2.6\%</math> when <math>n=100</math> and to a totally unacceptable <math>7\%</math> when <math>n=10</math>.  However, a very simple expedient of replacing <math>x</math> by  

If ''F'' is continuous then under the null hypothesis <math>\sqrt{n}D_n</math> converges to the Kolmogorov distribution, which does not depend on ''F''. This result may also be known as the Kolmogorov theorem. The accuracy of this limit as an approximation to the exact cdf of <math>K</math> when <math>n</math> is finite is not very impressive: even when <math>n=1000</math>, the corresponding maximum error is about <math>0.9\%</math>; this error increases to <math>2.6\%</math> when <math>n=100</math> and to a totally unacceptable <math>7\%</math> when <math>n=10</math>.  However, a very simple expedient of replacing <math>x</math> by  

如果F是连续的，则在原假设{\displaystyle {\sqrt {n}}D_{n}}下收敛到不依赖于F的Kolmogorov分布。该结果也称为Kolmogorov定理。当n为有限时，此极限的精确度近似为K的确切累积分布函数，效果并不十分令人满意：即使n = 1000，相应的最大误差约为0.9%。此错误在100时增加到2.6%，在10时增加到完全不可接受的7%。但是，如果将x替换为

如果F是连续的，则在原假设{\displaystyle {\sqrt {n}}D_{n}}下收敛到不依赖于F的Kolmogorov分布。该结果也称为Kolmogorov定理。当n为有限时，此极限的精确度近似为K的确切累积分布函数，效果并不十分令人满意：即使n = 1000，相应的最大误差约为0.9%。此错误在100时增加到2.6%，在10时增加到完全不可接受的7%。但是，如果将x替换为

in the argument of the Jacobi theta function reduces these errors to  

in the argument of the Jacobi theta function reduces these errors to <math>0.003\%</math>, <math>0.027\%</math>, and <math>0.27\%</math> respectively; such accuracy would be usually considered more than adequate for all practical applications.

<math>0.003\%</math>, <math>0.027\%</math>, and <math>0.27\%</math> respectively; such accuracy would be usually considered more than adequate for all practical applications.

在Jacobi theta函数的参数e中，将这些误差分别减小到0.003％，0.027％和0.27％；该精度足以满足现阶段所有实际应用，。

The ''goodness-of-fit'' test or the Kolmogorov–Smirnov test can be constructed by using the critical values of the Kolmogorov distribution. This test is asymptotically valid when <math>n \to\infty</math>. It rejects the null hypothesis at level <math>\alpha</math> if

The ''goodness-of-fit'' test or the Kolmogorov–Smirnov test can be constructed by using the critical values of the Kolmogorov distribution. This test is asymptotically valid when <math>n \to\infty</math>. It rejects the null hypothesis at level <math>\alpha</math> if

where ''K''_''α'' is found from

where ''K''_''α'' is found from

The asymptotic [[statistical power|power]] of this test is 1.

The asymptotic [[statistical power|power]] of this test is 1.

该渐进检测效能为1。

该渐进检测效能为1。

用于计算任意n和x的累积分布函数{\displaystyle \operatorname {Pr} (D_{n}\leq x)}或其补数的快速准确的算法:

用于计算任意n和x的累积分布函数<math>\operatorname{Pr}(D_n \leq x)</math>或其补数的快速准确的算法:

• 统计软件期刊2011年Journal of Statistical Software刊登的Simard R, L'Ecuyer P的文章《计算双向Kolmogorov–Smirnov分布》以及统计与概率通信期刊2017年刊登的Moscovich A, Nadler B 的文章《快速计算泊松过程的边界穿越概率》。在文章《计算双向Kolmogorov–Smirnov分布》中找到具有C和Java代码的连续零分布。

• 统计软件期刊2011年Journal of Statistical Software刊登的Simard R, L'Ecuyer P的文章《计算双向Kolmogorov–Smirnov分布》以及统计与概率通信期刊2017年刊登的Moscovich A, Nadler B 的文章《快速计算泊松过程的边界穿越概率》。在文章《计算双向Kolmogorov–Smirnov分布》中找到具有C和Java代码的连续零分布。

• 统计软件期刊2019年Journal of Statistical Software刊登的Dimitrova DS, Kaishev VK, Tan S的文章《当潜在累积分布函数是完全离散，混合或连续时，计算Kolmogorov–Smirnov分布》和Dimitrova, Dimitrina; Kaishev, Vladimir; Tan, Senren.的文章《KSgeneral：计算（离散）连续零分布的K-S检验的P值》。对于R项目的KSgeneral软件包中实现的纯离散，混合或连续零分布，可以进行统计计算，对于给定的样本，它还可以计算KS检验统计量及其p值。或者，可以从文章《当潜在累积分布函数是完全离散，混合或连续时，计算Kolmogorov–Smirnov分布》中获得替代的C ++实现。

• 统计软件期刊2019年Journal of Statistical Software刊登的Dimitrova DS, Kaishev VK, Tan S的文章《当潜在累积分布函数是完全离散，混合或连续时，计算Kolmogorov–Smirnov分布》和Dimitrova, Dimitrina; Kaishev, Vladimir; Tan, Senren.的文章《KSgeneral：计算（离散）连续零分布的K-S检验的P值》。对于R项目的KSgeneral软件包中实现的纯离散，混合或连续零分布，可以进行统计计算，对于给定的样本，它还可以计算KS检验统计量及其p值。或者，可以从文章《当潜在累积分布函数是完全离散，混合或连续时，计算Kolmogorov–Smirnov分布》中获得替代的C ++实现。