A comparison of some test statistics of the Kolmogorov type

Abstract

It is natural to associate optimal estimation and optimal statistical testing. In this paper continuous function estimation of the cumulative distribution is used to define two test statistics that compete with the Kolmogorov Dn statistic. The first statistic, Cn, is attributed to Pyke and the second, Rn, is obtained by polygonalizing the sample distribution function. It is known that both are asymptotically equivalent to the Kolmogorov statistic. Using the methods of J. Durbin, the small sample distributions are tabled as well as the critical points for significance levels of .20, .10 , .05 , .025 , and .01. It is shown that is stochastically smaller than Dn and it appears that Cn is also smaller than Dn.