A Saddlepoint Approximation to the Limiting Distribution of a K-sample Baumgartner Statistic

Testing hypothesis is one of the most important problems in a nonparametric statistic. Various nonparametric test statistics have been proposed and discussed for a long time. We use the exact critical value for testing hypothesis when the sample sizes are small. However, for large sample sizes, it is very difficult to evaluate the exact critical value. Therefore, the limiting distributions of nonparametric tests are needed for testing the hypothesis. The purpose of this paper is to derive the limiting distribution of a k-sample Baumgartner test proposed by Murakami (2006). At first we consider a two-sample problem, which is one of the most common types of statistical problems. Let X = (X1, . . . , Xn) and Y = (Y1, . . . , Ym) be two random samples of size n and m independent observations, each of which has a continuous distribution described as F (x) and G(y), respectively. Let R1 < · · · < Rn and H1 < · · · < Hm denote the combined-samples ranks of the X-value and Y -value in increasing order of magnitude, respectively. One of the problems is to test the hypothesis H0 : F = G against H1 : not H0. Baumgartner et al. (1998) defined a novel nonparametric two-sample statistic for the hypothesis as