On the Range of the Difference between Hypothetical Distribution Function and Pyke's Modified Empirical Distribution Function

Abstract

The statistic described in the title may be used to provide a test of the hypothesis that a population has a prescribed continuous distribution function. It differs from that proposed by Kuiper for use with distributions on a circle only in that the usual empirical distribution function is replaced by Pyke's modification. This change and a theorem of Sparre Andersen make possible a computation of the exact distribution for finite sample size along the lines of the computation of the distribution of Kolmogorov's statistic. A brief comparison is made in Section 1 of this statistic, Kolmogorov's, and a statistic studied by Sherman, as distances between hypothetical and empirical distribution functions. The asymptotic distribution, due to Darling and Kuiper, appears in Section 3. Tables are included of the distribution for sample sizes 1 through 20.