On integer partitions and the Wilcoxon rank-sum statistic

Andrew V. Sills
{"title":"On integer partitions and the Wilcoxon rank-sum statistic","authors":"Andrew V. Sills","doi":"arxiv-2409.05741","DOIUrl":null,"url":null,"abstract":"In the literature, derivations of exact null distributions of rank-sum\nstatistics is often avoided in cases where one or more ties exist in the data.\nBy deriving the null distribution in the no-ties case with the aid of classical\n$q$-series results of Euler and Rothe, we demonstrate how a natural\ngeneralization of the method may be employed to derive exact null distributions\neven when one or more ties are present in the data. It is suggested that this\nmethod could be implemented in a computer algebra system, or even a more\nprimitive computer language, so that the normal approximation need not be\nemployed in the case of small sample sizes, when it is less likely to be very\naccurate. Several algorithms for determining exact distributions of the\nrank-sum statistic (possibly with ties) have been given in the literature (see\nStreitberg and R\\\"ohmel (1986) and Marx et al. (2016)), but none seem as simple\nas the procedure discussed here which amounts to multiplying out a certain\npolynomial, extracting coefficients, and finally dividing by a binomal\ncoefficient.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"41 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - STAT - Statistics Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05741","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

In the literature, derivations of exact null distributions of rank-sum statistics is often avoided in cases where one or more ties exist in the data. By deriving the null distribution in the no-ties case with the aid of classical $q$-series results of Euler and Rothe, we demonstrate how a natural generalization of the method may be employed to derive exact null distributions even when one or more ties are present in the data. It is suggested that this method could be implemented in a computer algebra system, or even a more primitive computer language, so that the normal approximation need not be employed in the case of small sample sizes, when it is less likely to be very accurate. Several algorithms for determining exact distributions of the rank-sum statistic (possibly with ties) have been given in the literature (see Streitberg and R\"ohmel (1986) and Marx et al. (2016)), but none seem as simple as the procedure discussed here which amounts to multiplying out a certain polynomial, extracting coefficients, and finally dividing by a binomal coefficient.
关于整数分区和威尔科克森秩和统计量
通过借助欧拉和罗特的经典 q$ 系列结果推导无并列情况下的零分布,我们展示了如何利用该方法的自然概括来推导精确的零分布,即使数据中存在一个或多个并列。我们建议这种方法可以在计算机代数系统,甚至更简单的计算机语言中实现,这样在样本量较小的情况下就不必使用正态近似,因为正态近似不太可能非常精确。文献中已经给出了几种确定柄和统计量精确分布(可能有并列关系)的算法(见 Streitberg and R\"ohmel (1986) and Marx et al. (2016)),但似乎都不如这里讨论的程序简单,它相当于乘出某个二项式,提取系数,最后除以二项式系数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信