柯尔莫可洛夫-斯米洛夫检验（K-S test）

    在统计学中，柯尔莫可洛夫-斯米洛夫检验基于累计分布函数，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。

     在进行累计概率（cumulative probability）统计的时候，你怎么知道组之间是否有显著性差异？有人首先想到单因素方差分析或双尾检验（2 tailed TEST）。其实这些是不准确的，最好采用Kolmogorov-Smirnov test（柯尔莫诺夫-斯米尔诺夫检验）来分析变量是否符合某种分布或比较两组之间有无显著性差异。

分类：

1、Single sample Kolmogorov-Smirnov goodness-of-fit hypothesis test.

      采用柯尔莫诺夫-斯米尔诺夫检验来分析变量是否符合某种分布，可以检验的分布有正态分布、均匀分布、Poission分布和指数分布。指令如下：

>> H = KSTEST(X,CDF,ALPHA,TAIL) % X为待检测样本，CDF可选：如果空缺，则默认为检测标准正态分布；

如果填写两列的矩阵，第一列是x的可能的值，第二列是相应的假设累计概率分布函数的值G(x)。ALPHA是显著性水平（默认0.05）。TAIL是表示检验的类型（默认unequal,不平衡）。还有larger，smaller可以选择。

 如果，H=1 则否定无效假设； H=0，不否定无效假设（在alpha水平上）

例如，

x = -2:1:4
x =
  -2  -1   0   1   2   3   4

[h,p,k,c] = kstest(x,[],0.05,0)
h =
   0
p =
   0.13632
k =
   0.41277
c =
   0.48342

The test fails to reject the null hypothesis that the values come from a standard normal distribution.

2、Two-sample Kolmogorov-Smirnov test

     检验两个数据向量之间的分布的。

>>[h,p,ks2stat] = kstest2(x1,x2,alpha,tail)

% x1,x2都为向量，ALPHA是显著性水平（默认0.05）。TAIL是表示检验的类型（默认unequal,不平衡）。

例如，x = -1:1:5
y = randn(20,1);
[h,p,k] = kstest2(x,y)
h =
     0
p =
    0.0774
k =
    0.5214         

wiki翻译起来太麻烦，还有可能曲解本意，最好看原版解释。

       In statistics, the Kolmogorov–Smirnov test (K–S test) is a form of minimum distance estimation used as a nonparametric test of equality of one-dimensional probability distributions used to compare a sample with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). The Kolmogorov–Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples. The null distribution of this statistic is calculated under the null hypothesis that the samples are drawn from the same distribution (in the two-sample case) or that the sample is drawn from the reference distribution (in the one-sample case). In each case, the distributions considered under the null hypothesis are continuous distributions but are otherwise unrestricted.

The two-sample KS test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.

The Kolmogorov–Smirnov test can be modified to serve as a goodness of fit test. In the special case of testing for normality of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using the sample to modify the null hypothesis reduces the power of a test. Correcting for this bias leads to the Lilliefors test. However, even Lilliefors' modification is less powerful than the Shapiro–Wilk test or Anderson–Darling test for testing normality.^[1]

Kolmogorov–Smirnov statistic

The empirical distribution function F_n for n iid observations X_i is defined as

http://upload.wikimedia.org/math/d/d/b/ddbf5a62e7d346dc072a2f47f1871530.pngtest" />

where http://upload.wikimedia.org/math/7/9/0/790f58e3dd7575eb52229c6b68d9c232.pngtest" /> is the indicator function, equal to 1 if X_i ≤ x and equal to 0 otherwise.

The Kolmogorov–Smirnov statistic for a given cumulative distribution function F(x) is

http://upload.wikimedia.org/math/3/b/8/3b8599f003f2a131d8084621b1c39640.pngtest" />

where sup S is the supremum of set S. By the Glivenko–Cantelli theorem, if the sample comes from distribution F(x), then D_n converges to 0 almost surely. Kolmogorov strengthened this result, by effectively providing the rate of this convergence (see below). The Donsker theorem provides yet stronger result.

Kolmogorov distribution

The Kolmogorov distribution is the distribution of the random variable

http://upload.wikimedia.org/math/1/b/7/1b7fd8f556e7382d973cb6bf95a245ea.pngtest" />

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

http://upload.wikimedia.org/math/2/8/9/2899bf257fc0aa1f48b3ffcff8f783ae.pngtest" />

Kolmogorov–Smirnov test

Under null hypothesis that the sample comes from the hypothesized distribution F(x),

http://upload.wikimedia.org/math/8/4/2/842d0b1d85ca11aa30ccc90a09936fa4.pngtest" />

in distribution, where B(t) is the Brownian bridge.

If F is continuous then under the null hypothesis http://upload.wikimedia.org/math/1/e/c/1ec425f3720cd63ffabd65504c798972.pngtest" /> converges to the Kolmogorov distribution, which does not depend on F. This result may also be known as the Kolmogorov theorem; see Kolmogorov's theorem for disambiguation.

The goodness-of-fit test or the Kolmogorov–Smirnov test is constructed by using the critical values of the Kolmogorov distribution.

The null hypothesis is rejected at level α if

http://upload.wikimedia.org/math/8/9/1/891bbf7487bdbedcc202cb47bee880ac.pngtest" />

where K_α is found from

http://upload.wikimedia.org/math/b/b/4/bb4772bb6ae01da6b6a3d1d6b3b43097.pngtest" />

The asymptotic power of this test is 1. If the form or parameters of F(x) are determined from the X_i, the inequality may not hold. In this case, Monte Carlo or other methods are required to determine the rejection level α.