A Procedure to Find Exact Critical Values of Kolmogorov-Smirnov Test

The compatibility of a random sample of data with a given distribution can be checked by a goodness of fit test. Among a number of proposals one of the most important was suggested by Kolmogorov (1933) and Smirnov (1939). They proposed the Dn statistic based on the comparison between the hypothesized distribution function F0(x) and the empirical distribution function of the sample Sn(x). If F0(x) is continuous and under the null hypothesis, the distribution of Dn is independent of F0(x), i.e. the test is distribution-free. In this paper a procedure providing the exact critical values of the Kolmogorov-Smirnov test for fixed significance levels is introduced. These values are obtained by a modification of the procedure proposed by Feller (1948). In particular, the distribution function of the test statistic is obtained by the solution of a linear system of equations whose coefficients are proper marginal and conditional probabilities.