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| − | The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to multivariate data. This is not straightforward because the maximum difference between two joint cumulative distribution functions is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of \Pr(x < X \land y < Y) or \Pr(X < x \land Y > y) or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.
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| − | 如果要对多变量数据应用类似的检验，则需要修改 Kolmogorov-Smirnov 检验统计量。这并不简单，因为两个联合累积分布函数的最大差值通常不等于任何一个互补分布函数的最大差值。因此，最大的差异将取决于 Pr (x < x 土地 y < y)或 Pr (x < x 土地 y > y)或其他两种可能的安排中的任何一种。有人可能会要求所使用的测试的结果不应该取决于作出的选择。
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| | | <math>c({\alpha})</math> || 1.073 || 1.138 || 1.224 || 1.358 || 1.48 || 1.628 || 1.731 || 1.949 | | | <math>c({\alpha})</math> || 1.073 || 1.138 || 1.224 || 1.358 || 1.48 || 1.628 || 1.731 || 1.949 |
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| − | One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K–S statistics. In d dimensions, there are 2d−1 such orderings. One such variation is due to Peacock
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| | The Kolmogorov-Smirnov test (one or two sampled test verifies the equality of distributions) is implemented in many software programs: | | The Kolmogorov-Smirnov test (one or two sampled test verifies the equality of distributions) is implemented in many software programs: |
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| − | Kolmogorov-Smirnov 检验(一个或两个抽样检验验证分布是否相等)在许多软件程序中实现:
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| − | Here, again, the larger the sample sizes, the more sensitive the minimal bound: For a given ratio of sample sizes (e.g. <math>m=n</math>), the minimal bound scales in the size of either of the samples according to its inverse square root.
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| − | 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. <ref>{{cite journal |last1=Marozzi |first1=Marco |title=Some Notes on the Location-Scale Cucconi Test |journal=Journal of Nonparametric Statistics |date=2009 |volume=21 |issue=5 |page=629–647 |doi=10.1080/10485250902952435 }}</ref> and <ref>{{cite journal |last1=Marozzi |first1=Marco |title=Nonparametric Simultaneous Tests for Location and Scale Testing: a Comparison of Several Methods |journal=Communications in Statistics – Simulation and Computation |date=2013 |volume=42 |issue=6 |page=1298–1317 |doi=10.1080/03610918.2012.665546 }}</ref> 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.
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| | ==Setting confidence limits for the shape of a distribution function== | | ==Setting confidence limits for the shape of a distribution function== |