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第38行： + + − − 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 第57行：
第57行： − :<math>D_n= \sup_x |F_n(x)-F(x)|</math>+ − + − where sup<sub>''x''</sub> is the [[supremum]] of the set of distances. By the [[Glivenko–Cantelli theorem]], if the sample comes from 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-Smirnov test#Kolmogorov distribution|Kolmogorov distribution]]). [[Donsker's theorem]] provides a yet stronger result. − − The Kolmogorov distribution is the distribution of the random variable − − 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)|+ − K = sup _ { t in [0,1]} | b (t) | + + + +
→Kolmogorov–Smirnov statistic Kolmogorov-Smirnov统计
== Kolmogorov–Smirnov statistic Kolmogorov-Smirnov统计==
== Kolmogorov–Smirnov statistic Kolmogorov-Smirnov统计==
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
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.
n个独立且均匀分布（i.i.d.）的有序观测值Xi的经验分布函数Fn定义为
F_{n}(x)={1 \over n}\sum _{i=1}^{n}I_{[-\infty ,x]}(X_{i})
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。
给定累积分布函数F（x）的Kolmogorov–Smirnov统计量为
D_{n}=\sup _{x}|F_{n}(x)-F(x)|
其中supx是距离集的最大值。根据Glivenko-Cantelli定理，如果样本来自分布F（x），则当n变为无穷大时，Dn几乎肯定会收敛于0。Kolmogorov通过有效加入收敛速率来增强此结果（请参阅Kolmogorov分布）。另外Donsker定理提供了更强的结果。
在实践中，该统计需要相对大量的数据点（与其他拟合优度标准相比，例如Anderson-Darling检验统计）才能正确地拒绝原假设。
==Kolmogorov distribution==
==Kolmogorov distribution==