Friday Ewere and Sunday Martins Ogbonmwan
Keywords: Wilcoxon Rank Sum Test, exact test, rank test, permutation test, combinatorics.
Abstract: Generating the entire permutation sample space especially when sample sizes are not small have been a major problem in constructing an exact test of significance of a rank statistic. Recently, the use of softwares for computing statistical tests has become common. However, procedures on this software for calculating the significance levels for many nonparametric tests are based on asymptotic results. These asymptotic results are only reliable when sample sizes are large enough. Unfortunately, the definition of what constitute a large sample size for most statistics is quite vague. The aim of this paper is to formulate a method for obtaining the exact distribution of a rank statistic. The proposed method is based on combinatorics in the representation of the probability generating function of the test statistic. The proposed method bypasses the problem of actually carrying out a complete enumeration in a permutation test. Essentially, the exact critical values for the Wilcoxon Rank Sum (WRS) test statistic are produced. The asymptotic property of the WRS is carefully studied and the minimum sample size required for the application of the large sample approximation is provided.
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