Kolmogorov-Smirnov tests when parameters are estimated with applications to tests of exponentiality and tests on spacings

Abstract

The paper considers Kolmogorov-Smirnov tests of goodness of fit in the presence of unknown nuisance parameters. It is shown that for a wide class of cases the acceptance probability is the ratio of two densities of which the denominator density is known. Methods of calculating and inverting the Fourier transform of the numerator density are considered. The results are applied to the case of an exponential distribution which has unknown mean and perhaps also unknown lower terminal. Tables of percentage points are given for the standard Kolmogorov-Smirnov statistics Dn-,Dn+ and Dn. It is shown that these tables also give the percentage points for analogous statistics derived from the sample distribution function of the spacings between uniform order statistics.