Exact Moments of Two-Sample Rank Statistics: Improving the Approximate Distribution and Applications to Lepage-Type Tests

Abstract The two-sample testing problem is a fundamental topic in various scientific fields, with numerous nonparametric test statistics proposed over the years. This study derives the exact expected value and variance of a two-sample rank statistic with arbitrary score functions. Utilizing the exact first two moments, an approximation to the distribution of various test statistics is considered for finite sample sizes. Furthermore, the exact expected value and variance are applied to a test statistic designed for testing of location and scale parameters. Monte Carlo simulation studies are conducted to evaluate the validity of the proposed approximation and to compare the power of the suggested test against existing methods. The simulation results demonstrate that the proposed statistic is a competitive alternative to established simultaneous test statistics. The effectiveness of our approach is demonstrated through applications to real dataset, and we conclude with a discussion of our findings and potential directions for future research.