17.5.5.2 Algorithms (Two-Sample Kolmogorov-Smirnov Test)KS-Test-Algorithm
The procedure below draws on NAG algorithms.
Consider two independent samples X and Y, with the size of /math-8d88a43872d6db6535d8672a15f09ce2.png?v=0 "n_1\,\!") and /math-ede03c8aef1b07c898c7747b489fd765.png?v=0 "n_2\,\!") .Denoted as /math-d79752c45b6e0288c460390908ff4157.png?v=0 "x_1,x_2,\ldots ,x_{n_1}\,\!") and /math-ba3bf26bc53cf72dd2f7e804e8351ba7.png?v=0 "y_1,y_2,\ldots ,y_{n_1}\,\!") respectively. Let F(x) and G(x) represent their respective, unknown distribution functions. Also let /math-2eaa8777a1f571cbad7bfca3adf83c34.png?v=0 "S_1(x)\,\!") and /math-17118f898c20f6dbf66463ce22714faa.png?v=0 "S_2(x)\,\!") denote the values of sample empirical distribution functions.
The null hypothesis :F(x)=G(x)
The alternative hypothesis /math-fabff59271b950125b7a360fba21de2c.png?v=0 "H_1\,\!") :F(x)<>G(x) the associated p-value is a two-tailed probability;
or :F(x)>G(x) the associated p-value is an upper-tailed probability,
or : F(x)<G(x) the associated p-value is a lower-tailed probability
For the first case of , the statistics /math-a9aca3b08301eb7213e2408c0ab85104.png?v=0 "D_{n_1,n_2} \,\!") represents the largest absolute deviation of the two empirical distribution functions.
For the second case of , the statistics /math-4cd60644c5463cc66c7631c167ed3936.png?v=0 "D_{n_1,n_2}^{+} \,\!") represents the largest positive deviation between the empirical distribution function of the first sample and the empirical distribution function of the second sample, that is /math-9cf98ed19b4ee4732ba5e4524e54a3f8.png?v=0 "D_{n_1,n_2}^{+}=\max \{S_1(x)-S_2(x),0\}\,\!") .
For the third case of , the statistics /math-8af9c1a0d88a260554650e06fd515607.png?v=0 "D_{n_1,n_2}^{-} \,\!") represents the largest positive deviation between the empirical distribution function of the second sample and the empirical distribution function of the first sample, that is /math-6a6a55b10a26fafb59197daab9f6d53b.png?v=0 "D_{n_1,n_2}^{-}=\max \{S_2(x)-S_1(x),0\}\,\!") .
KS-test2 also returns the standard statistics /math-3e29b728262edf3639e74270b72e638a.png?v=0 "Z=\sqrt{(n_1*n_2)/(n_1+n_2)}*D\,\!") ,
where /math-a7c6c783c5d03fc91d0594b217f56580.png?v=0 "D\,\!") maybe /math-3f75caf92b4b313ec5ec353b2df6a0b0.png?v=0 "D_{n_1,n_2}\,\!") ,, depending on the choice of the alternative hypothesis.
The distribution of the statistic /math-07b1048bc60901fe82394ae47282671c.png?v=0 "Z\,\!") converges asymptotically to a distribution given by Smirnov as and /math-2f6fac59fce80e76d698bcf5ea77bab6.png?v=0 "n_2\,\!") increase. The probability, under the null hypothesis, of obtaining a value of the test statistic as extreme as that observed, is computed.
If /math-caeb72e22c815c3c25d6a8b427c8aaad.png?v=0 "max(n_1,n_2)\leq 2500\,\!") and /math-be2983946ebf66a1fd9de27574f27b22.png?v=0 "n_1*n_2\leq 10000\,\!") then an exact method is given by Kim and Jinrich. Otherwise /math-c0f582773fdbd168bbab09a1e6159c46.png?v=0 "p\,\!") is computed using the approximations suggested by Kim and Jenrich (1973)
Note that the method used only exact for continuous theoretical distributions.
This method computes the two-sided probability. The one-sided probabilities are estimated by having the two-sided probability. This is a good estimate for small , that is /math-cd60a65bcbb8b47dc37b5c879ed79777.png?v=0 "p\leq 0.10\,\!") , but it becomes very poor for larger .
For more details of the algorithm, please refer to nag_2_sample_ks_test (g08cdc) .
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