The Kolmogorov-Smirnov Test

One-Sample Test

This test is another goodness-of-fit test: Given a sample $(X_1,\,...,\,X_n)}$ , we would like to test $H_{0}:\,{distribution} \; F}\;/\;H_1:\,{distribution} <> F}$ . By the Glivenko-Cantelli theorem, the empirical distribution function

$F_n(x) := \frac{|\{i <= n | X_i <= x\}|}n$

converges uniformly to the actual distribution function of . So we choose

$D_n := \Vert F_n - F\Vert = \sup_{x} | F_n(x) - F(x)|$

as our test statistic, which should be small (under ).

One can prove that for continuous , the null distribution of does not depend on , and can be calculated as follows:

$D_n = \max_{i=1}^n ( \max ( | F(X_{n......ft| F(X_{n:i}) - \frac{i-1}n | ) )$

where $X_{n:1} < X_{n:2} < ... < X_{n:n}$ denote the order statistics.

One can even calculated the asymptotic distribution function for large : Let $lambda_n := \sqrtn * D_n$ . Then, for ,

$\P [lambda_n <= x] –> K(x) := \{\begin{array}{l...... & {if} \; x > 0 \\\end{array} .$