Confidence Intervals for Probabilities and Tolerance Regions Based on a Generalization of the Mann-Whitney Statistic

Abstract Inference procedures are proposed for any specified function of two random variables f(X, Y), assuming that independent random samples from the X and Y populations are available. A generalization of the Mann-Whitney statistic is used to obtain point and interval estimates for the probability that f(X, Y) falls in a given interval. One proposed confidence interval is a modification and improvement to the approach of Halperin, Gilbert, and Lachin (1987) for estimating Pr(X < Y). A thorough review is made of the literature on nonparametric confidence bounds for Pr(X < Y), with emphasis on methods based on the central limit theorem. In addition to the improvement of Halperin et al.'s method, bounds analogous to the Clopper-Pearson confidence interval for a binomial parameter are proposed. Simulation results show the performance of the alternative procedures. Approximate nonparametric tolerance limits for f(X, Y) are also proposed. The article closes by constructing a two-sided tolerance interval for ...