Evaluating Kolmogorov's Distribution

Abstract

Kolmogorov's goodness-of-fit measure, D_n , for a sample CDF has consistently been set aside for methods such as the D⁺_n or D^-_n of Smirnov, primarily, it seems, because of the difficulty of computing the distribution of D_n . As far as we know, no easy way to compute that distribution has ever been provided in the 70+ years since Kolmogorov's fundamental paper. We provide one here, a C procedure that provides Pr(D_n < d) with 13-15 digit accuracy for n ranging from 2 to at least 16000. We assess the (rather slow) approach to limiting form, and because computing time can become excessive for probabilities>.999 with n's of several thousand, we provide a quick approximation that gives accuracy to the 7th digit for such cases.