## Kolmogorov-Smirnov test

$F_N (x)$ is the empirical distribution function of $N$ generated pseudo-random number sequence $x_0,x_1,\ldots ,x_{N-1}$.

$F_N(x)=\frac{1}{N}\left\{The number of random numbers that satisfy x_i\leq x\right\}$
(1)

Whether the discrepancy of the distribution function $F_0(x)$ of the assumed probability distribution is beyond the range of variation is tested. When $F(x)$ is the population distribution function, the null hypothesis $H_0$ of the test (two-tailed test) is as follows,

$H_0:F(x)=F_0(x)$
(2)

$H_1:F(x)\neq F_0(x)$

Against the alternative hypothesis $H_1$, the test statistics is as follows,

$D=\underset{x}{\sup } \left|F_N(x)-F_0(x)\right|$

In comparison with the critical value determined by $N$ and the significance level $\alpha$, the test statistics is larger than that , it is determined the distribution of the statistical population is different from the distribution of the hypothesis. Critical value is defined by the following approximate expression,

$K_N(\alpha)=\sqrt{-0.5\log(\alpha/2)}$

References :
• M.Fushimi, Random number , UP Sensho Applied Mathematics , University of Tokyo Press , 1989
• Y.Tanaka,T.Tarumi, Statistical analysis handbook , Kyoritsu Shuppan , 1999

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