查看“︁柯尔莫哥洛夫-斯米尔诺夫检验”︁的源代码

{{NoteTA
|T=zh-cn:柯尔莫哥洛夫-斯米尔诺夫检验;zh-tw:科摩哥洛夫-史密諾夫檢定
|G1=Math
|1=zh-cn:柯尔莫哥洛夫-斯米尔诺夫检验;zh-tw:K-S檢定
|2=zh-cn:柯尔莫哥洛夫;zh-tw:科摩哥洛夫
|3=zh-cn:斯米尔诺夫;zh-tw:史密諾夫
|4=zh-tw:母數;zh-cn:参数;zh-hant:參數
}}
'''-{zh-cn:柯尔莫哥洛夫-斯米尔诺夫检验;zh-tw:科摩哥洛夫-史密諾夫檢定'''，簡稱'''K-S檢定}-'''（{{lang-en|Kolmogorov-Smirnov test}}，簡稱{{lang|en|K-S test}}），是一种基于累计分布函数的非参数检验，用以检验两个经验分布是否不同或一个经验分布与另一个理想分布是否不同。本檢定以[[安德雷·柯尔莫哥洛夫]]和{{le|尼古拉·斯米尔诺夫|Nikolai Smirnov (mathematician)}}之名作命名。

== 柯尔莫哥洛夫分布 ==
柯尔莫哥洛夫分布（kolmogorov distribution）是[[随机变量]]

:<math>K=\sup_{t\in[0,1]}|B(t)|,</math>

的分布，其中 <math>B(t)</math> 是[[布朗桥]]。K的[[累积分布函数]]由下式给出

:<math>\operatorname{Pr}(K\leq x)=1-2\sum_{i=1}^\infty (-1)^{i-1} e^{-2i^2 x^2}=\frac{\sqrt{2\pi}}{x}\sum_{i=1}^\infty e^{-(2i-1)^2\pi^2/(8x^2)}.</math>

柯尔莫哥洛夫-斯米尔诺夫检验的统计量形式及其在零假设下的渐近分布是由[[安德雷·柯尔莫哥洛夫]]<ref name=AK>{{Cite journal |author=Kolmogorov A |year=1933 |title=Sulla determinazione empirica di una legge di distribuzione |journal=G. Inst. Ital. Attuari |volume=4 |pages=83}}</ref>提出的。

== 参考文献 ==
*Justel, A., Peña, D. and Zamar, R. (1997) ''A multivariate Kolmogorov-Smirnov test of goodness of fit'', Statistics & Probability Letters, 35(3), 251-259. 
* {{cite book
  | last = Eadie
  | first = W.T.
  | coauthors = D. Drijard, F.E. James, M. Roos and B. Sadoulet
  | title = Statistical Methods in Experimental Physics
  | url = https://archive.org/details/statisticalmetho0000unse
  | publisher = North-Holland
  | year = 1971
  | location = Amsterdam
  | pages = [https://archive.org/details/statisticalmetho0000unse/page/269 269]–271
  | isbn = 0-444-10117-9 }}
* {{cite book
  | last1 = Stuart
  | first1 = Alan
  | first2 = Keith
  | last2 = Ord 
  | first3=Steven [F.]
  | last3=Arnold
  | title=Classical Inference and the Linear Model
  | edition=Sixth 
  | series = Kendall's Advanced Theory of Statistics
  | volume = 2A
  | year = 1999
  | publisher = Arnold, Oxford University Press
  | location = London, New York 
  | isbn=0-340-66230-1 
  | mr=1687411
  | pages = 25.37–25.43 }}
*Corder, G.W., Foreman, D.I. (2009).''Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach'' Wiley, ISBN 978-0-470-45461-9
*Stephens, M.A. (1979) ''Test of fit for the logistic distribution based on the empirical distribution function'', Biometrika, 66(3), 591-5.
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== 外部連結 ==
* {{springer|title=Kolmogorov-Smirnov test|id=p/k055740}}
* [http://www.physics.csbsju.edu/stats/KS-test.html Short introduction] {{Wayback|url=http://www.physics.csbsju.edu/stats/KS-test.html |date=20050710021649 }}
* [http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm KS test explanation] {{Wayback|url=http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm |date=20100114215353 }}
* [http://www.ciphersbyritter.com/JAVASCRP/NORMCHIK.HTM JavaScript implementation of one- and two-sided tests] {{Wayback|url=http://www.ciphersbyritter.com/JAVASCRP/NORMCHIK.HTM |date=20100110015231 }}
* [http://jumk.de/statistic-calculator/ Online calculator with the K-S test] {{Wayback|url=http://jumk.de/statistic-calculator/ |date=20100107005037 }}
* Open-source C++ code to compute the [http://root.cern.ch/root/html/TMath.html#TMath:KolmogorovProb Kolmogorov distribution] {{Wayback|url=http://root.cern.ch/root/html/TMath.html#TMath:KolmogorovProb |date=20091228030630 }} and perform the [http://root.cern.ch/root/html/TMath.html#TMath:KolmogorovTest K-S test] {{Wayback|url=http://root.cern.ch/root/html/TMath.html#TMath:KolmogorovTest |date=20091228030630 }}
* Paper on [http://www.jstatsoft.org/v08/i18/paper Evaluating Kolmogorov’s Distribution] {{Wayback|url=http://www.jstatsoft.org/v08/i18/paper |date=20150819090919 }}; contains C implementation. This is the method used in [[Matlab]].

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[[Category:常態性檢定]]
[[Category:統計距離]]