{"title":"反正弦分布下Wilcoxon和Student两样本检验样本量的实验比较","authors":"F. M. Al-athari","doi":"10.55463/issn.1674-2974.50.1.12","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":15926,"journal":{"name":"湖南大学学报(自然科学版)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Experimental Comparison of the Sample Sizes of the Two-Sample Tests of Wilcoxon and Student under the Arcsine Distribution\",\"authors\":\"F. M. Al-athari\",\"doi\":\"10.55463/issn.1674-2974.50.1.12\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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.\",\"PeriodicalId\":15926,\"journal\":{\"name\":\"湖南大学学报(自然科学版)\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-02-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"湖南大学学报(自然科学版)\",\"FirstCategoryId\":\"1087\",\"ListUrlMain\":\"https://doi.org/10.55463/issn.1674-2974.50.1.12\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"湖南大学学报(自然科学版)","FirstCategoryId":"1087","ListUrlMain":"https://doi.org/10.55463/issn.1674-2974.50.1.12","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.