Exact Power of Goodness-of-Fit Tests of Kolmogorov Type for Discontinuous Distributions

Abstract

Goodness-of-fit tests of Kolmogorov type reject a null hypothesis H 0 : F ( x ) = F *( x ) whenever the graph of the sample cumulative distribution function crosses one of two boundary functions, G 1 ( x ), G 2 ( x ). The best-known example of a test of this type is the Kolmogorov—Smirnov test D . When the true cumulative distribution function F ( x ) is continuous, a number of algorithms are available for calculating the exact powers of such tests. In this article it is shown that such algorithms can also be used to calculate the exact power and level of significance of Kolmogorov-type goodness-of-fit tests when F ( x ) is discontinuous.