Experimental Comparison of the Sample Sizes of the Two-Sample Tests of Wilcoxon and Student under the Arcsine Distribution

When statistical experiments are performed, a sample size should be chosen in some optimum way so that we should use a sample size no larger than necessary. This paper compares the minimum sample sizes of two-sample t-test and the Wilcoxon rank-sum test under the arcsine distribution, based on their power [1, 2]. To accomplish this task, some essential probabilities that are useful in the power and sample size determination of the Wilcoxon rank-sum test were derived by the author for computing the approximated formula given by Lehmann [2]. The composite numerical integration algorithm is used to compute these probabilities, which are related to the arcsine distribution. In this study, a computer program was built by the author to find the exact (simulated) minimum sample sizes n for any significant level and power by iterating on n with starting points for n provided by the approximated formulas of [2, 3]. The scientific novelty of this research paper is determining the minimum sample sizes by considering a new set of the arcsine distribution shift alternatives of the forms giving the left-hand endpoint of the displaced distribution as the quantile of order p, 0 < p < 1, of the second distribution rather than using alternatives that specify as the quantile of order p, p < 0.5. As considered by [1], a choice that prevents losing some important alternative hypotheses is an extension to the set of alternative hypotheses considered by Guenther [1]. The exact (simulated) minimum sample sizes were computed and compared with each other and with the corresponding approximated formulas given by Lehmann [2] and Guenther [3]. Numerical results showed that the approximated formulas are very accurate and the Wilcoxon rank-sum test is more efficient when the sample size is more than 45. Otherwise, the Student two-sample t-test is better.