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→Two-sample Kolmogorov–Smirnov test 双样本Kolmogorov–Smirnov检验
The Kolmogorov–Smirnov test may also be used to test whether two underlying one-dimensional probability distributions differ. In this case, the Kolmogorov–Smirnov statistic is
The Kolmogorov–Smirnov test may also be used to test whether two underlying one-dimensional probability distributions differ. In this case, the Kolmogorov–Smirnov statistic is
Kolmogorov–Smirnov检验也可用于检验两个基本的一维概率分布是否不同。在这种情况下，Kolmogorov-Smirnov统计量为
<math>D_{n,m}=\sup_x |F_{1,n}(x)-F_{2,m}(x)|,</math>
<math>D_{n,m}=\sup_x |F_{1,n}(x)-F_{2,m}(x)|,</math>
where <math>F_{1,n}</math> and <math>F_{2,m}</math> are the [[empirical distribution function]]s of the first and the second sample respectively, and <math>\sup</math> is the [[Infimum and supremum|supremum function]].
where <math>F_{1,n}</math> and <math>F_{2,m}</math> are the [[empirical distribution function]]s of the first and the second sample respectively, and <math>\sup</math> is the [[Infimum and supremum|supremum function]].
其中F1，n和F2，m分别是第一样本和第二样本的经验分布函数，而sup是最高函数。对于量大的样本，如果满足以下条件，则原假设在α级被拒绝：
For large samples, the null hypothesis is rejected at level <math>\alpha</math> if
For large samples, the null hypothesis is rejected at level <math>\alpha</math> if
对于量大的样本，如果满足以下条件，则原假设在α级被拒绝
:<math>D_{n,m}>c(\alpha)\sqrt{\frac{n + m}{n\cdot m}}.</math>
:<math>D_{n,m}>c(\alpha)\sqrt{\frac{n + m}{n\cdot m}}.</math>
Where <math>n</math> and <math>m</math> are the sizes of first and second sample respectively. The value of <math>c({\alpha})</math> is given in the table below for the most common levels of <math>\alpha</math>
Where <math>n</math> and <math>m</math> are the sizes of first and second sample respectively. The value of <math>c({\alpha})</math> is given in the table below for the most common levels of <math>\alpha</math>
其中n和m分别是第一样本和第二样本的大小。下表给出了在α级最常见的α值c（α）：
| <math>\alpha</math> || 0.20 || 0.15 || 0.10 || 0.05 || 0.025 || 0.01 || 0.005 || 0.001
| <math>\alpha</math> || 0.20 || 0.15 || 0.10 || 0.05 || 0.025 || 0.01 || 0.005 || 0.001
and in general by
and in general by
一般而言：
:<math>c\left(\alpha\right)=\sqrt{-\ln\left(\tfrac{\alpha}{2}\right)\cdot \tfrac{1}{2}},</math>
:<math>c\left(\alpha\right)=\sqrt{-\ln\left(\tfrac{\alpha}{2}\right)\cdot \tfrac{1}{2}},</math>
so that the condition reads
so that the condition reads
以至于条件为：
:<math>D_{n,m}>\frac{1}{\sqrt{n}}\cdot\sqrt{-\ln\left(\tfrac{\alpha}{2}\right)\cdot \tfrac{1 + \tfrac{n}{m}}{2}}.</math>
:<math>D_{n,m}>\frac{1}{\sqrt{n}}\cdot\sqrt{-\ln\left(\tfrac{\alpha}{2}\right)\cdot \tfrac{1 + \tfrac{n}{m}}{2}}.</math>
Here, again, the larger the sample sizes, the more sensitive the minimal bound: For a given ratio of sample sizes (e.g. m=n), the minimal bound scales in the size of either of the samples according to its inverse square root.
Here, again, the larger the sample sizes, the more sensitive the minimal bound: For a given ratio of sample sizes (e.g. m=n), the minimal bound scales in the size of either of the samples according to its inverse square root.
同样，样本量越大，最小界限越敏感：对于给定比率的样本大小（例如m = n），最小界限根据其平方根的倒数来缩放两个样本的大小。
Note that the two-sample test checks whether the two data samples come from the same distribution. This does not specify what that common distribution is (e.g. whether it's normal or not normal). Again, tables of critical values have been published. A shortcoming of the Kolmogorov–Smirnov test is that it is not very powerful because it is devised to be sensitive against all possible types of differences between two distribution functions. and showed evidence that the Cucconi test, originally proposed for simultaneously comparing location and scale, is much more powerful than the Kolmogorov–Smirnov test when comparing two distribution functions.
Note that the two-sample test checks whether the two data samples come from the same distribution. This does not specify what that common distribution is (e.g. whether it's normal or not normal). Again, tables of critical values have been published. A shortcoming of the Kolmogorov–Smirnov test is that it is not very powerful because it is devised to be sensitive against all possible types of differences between two distribution functions. and showed evidence that the Cucconi test, originally proposed for simultaneously comparing location and scale, is much more powerful than the Kolmogorov–Smirnov test when comparing two distribution functions.
这里要注意的是两个样本检验出来的数据样本是否来自同一分布。其并未指定该共同分布是什么（例如，它是正常还是不正常）。而且关键值表已经得出。Kolmogorov–Smirnov检验没有那么有效，因为它被设计为对两个分布函数之间所有可能的差异敏感。如刊登在Journal of Nonparametric Statistics2009年刊上Marozzi, Marco (2009)的文章《Some Notes on the Location-Scale Cucconi Test》和刊登在Communications in Statistics – Simulation and Computation2013年刊上同样Marozzi, Marco (2009)的文章《Nonparametric Simultaneous Tests for Location and Scale Testing: a Comparison of Several Methods显示了证据，当比较两个分布函数时，最初建议同时比较位置和比例的Cucconi检验比Kolmogorov-Smirnov检验更有效。
==Setting confidence limits for the shape of a distribution function==
==Setting confidence limits for the shape of a distribution function==