Tests of the Kolmogorov-Smirnov Type for Exponential Data with Unknown Scale, and Related Problems

Abstract

Let D^ , D^+ and D^− denote Kolmogorov-Smirnov typo one-sample statistics to test good ness of fit in the presence of unknown nuisance parameters; then the distributions of D^ , D^+ and D^− depend on the population sampled and the estimator used. Simulation has been the primary tool for studying these statistics. Recently, Durbin obtained the distributions of D^ , D^+ and D^− in terms of a Fourier transform for a wide class of underlying populations, and produced explicit results for the exponential case. In this paper, the distribution functions of D^ , D^+ and D^− for the exponential case are derived from general results for order statistics, and computationally efficient approximations to these distribution functions are obtained. In the course of this derivation, Bonferroni inequalities of Kounias, and Sobel & Uppuluri are generalized. Certain problems of goodness-of-fit testing in the presence of nuisance parameters, whose solutions make use of existing tables, are also discussed. These problems include the Pareto, Rayleigh, power function, and uniform distributions.