Bernhard Kabelka - Mathematical Statistics: "The Kolmogorov-Smirnov Test

Two-Sample Test

If we would like to compare the two samples $(X_1,\,...,\,X_n)}$ and $(Y_1,\,...,\,Y_m)$ and would like to know whether the underlying distributions are the same, we may use the test statistic

$D_{n,m} := \Vert F_n - G_m\Vert$

where and resp. are the empirical distribution functions. Again, for continuous , the null distribution of $D_{n,m}$ does not depend on . $D_{n,m}$ may be calculated as follows:

$D_{n,m} = \max ( \max_{i <= n} | F_n(X_{n:......| F_n(Y_{m:i}) - G_m(Y_{m:i}) | )$

Finally, $lambda_n := \sqrt{\frac{nm}{n+m} * D_{n,m}$ has an asymptotic distribution function .