{"title":"为支持均匀分布而获得覆盖所需的观测数的分布","authors":"R. N. Rattihalli","doi":"10.1080/07474946.2023.2242414","DOIUrl":null,"url":null,"abstract":"AbstractFor a given positive number ‘δ′, we consider a sequence of δ− neighborhoods of the independent and identically distributed (i.i.d.) random variables, from a U(0,1) distribution, and “stop as soon as their union contains the interval (0,1).” We call such a union “a cover.” To find the distributions of N(δ), the stopping time random variable, we need the joint distribution of order statistics from a U(0,1) distribution. For each δ>0 and n=1,2,…, we obtain a general expression for P(N(δ)≤n), and for a fixed value of δ, it is the distribution function of N(δ). For a given n, let Δ(n) be the minimum value of δ, so that the union of the n δ− neighborhoods of the first n observations contains the interval (0,1). Because N(δ)≤n if and only if Δ(n)≤δ, the distributions of Δ(n) can be obtained by fixing n in the general expression for P(N(δ)≤n). To describe the impact of δ on the distribution of N(δ) and that of n on Δ(n), we sketch the graphs of distribution functions and the empirical distribution functions.Keywords: Neighborhoodstopping ruleuniform distributionSubject Classification: MSC 2020: 26B1562E15 ACKNOWLEDGMENTThanks to S. B. Patil and S. R. Rattihalli for their computational assistance.DISCLOSUREThe author has no conflicts of interest to report.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"4 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Distribution of number of observations required to obtain a cover for the support of a uniform distribution\",\"authors\":\"R. N. Rattihalli\",\"doi\":\"10.1080/07474946.2023.2242414\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"AbstractFor a given positive number ‘δ′, we consider a sequence of δ− neighborhoods of the independent and identically distributed (i.i.d.) random variables, from a U(0,1) distribution, and “stop as soon as their union contains the interval (0,1).” We call such a union “a cover.” To find the distributions of N(δ), the stopping time random variable, we need the joint distribution of order statistics from a U(0,1) distribution. For each δ>0 and n=1,2,…, we obtain a general expression for P(N(δ)≤n), and for a fixed value of δ, it is the distribution function of N(δ). For a given n, let Δ(n) be the minimum value of δ, so that the union of the n δ− neighborhoods of the first n observations contains the interval (0,1). Because N(δ)≤n if and only if Δ(n)≤δ, the distributions of Δ(n) can be obtained by fixing n in the general expression for P(N(δ)≤n). To describe the impact of δ on the distribution of N(δ) and that of n on Δ(n), we sketch the graphs of distribution functions and the empirical distribution functions.Keywords: Neighborhoodstopping ruleuniform distributionSubject Classification: MSC 2020: 26B1562E15 ACKNOWLEDGMENTThanks to S. B. Patil and S. R. Rattihalli for their computational assistance.DISCLOSUREThe author has no conflicts of interest to report.\",\"PeriodicalId\":48879,\"journal\":{\"name\":\"Sequential Analysis-Design Methods and Applications\",\"volume\":\"4 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-09-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Sequential Analysis-Design Methods and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1080/07474946.2023.2242414\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Sequential Analysis-Design Methods and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1080/07474946.2023.2242414","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
摘要
摘要对于给定的正数“δ”,我们从U(0,1)分布中考虑独立同分布(i.i.d)随机变量的δ -邻域序列,并且“当它们的并集包含区间(0,1)时就停止”。我们称这样的联盟为“掩护”。为了求出停止时间随机变量N(δ)的分布,我们需要U(0,1)分布的阶统计量的联合分布。对于每个δ>0且n=1,2,…,我们得到了P(n (δ)≤n)的一般表达式,对于固定值的δ,它是n (δ)的分布函数。对于给定的n,设Δ(n)为Δ的最小值,使得前n个观测值的n个Δ−邻域的并集包含区间(0,1)。因为N(δ)≤N当且仅当Δ(N)≤δ,所以将N固定在P(N(δ)≤N)的一般表达式中,即可得到Δ(N)的分布。为了描述δ对N(δ)和N对Δ(N)分布的影响,我们绘制了分布函数图和经验分布函数图。关键词:邻域停止规则均匀分布主题分类:MSC 2020: 26B1562E15感谢S. B. Patil和S. R. Rattihalli的计算协助。作者无利益冲突需要报告。
Distribution of number of observations required to obtain a cover for the support of a uniform distribution
AbstractFor a given positive number ‘δ′, we consider a sequence of δ− neighborhoods of the independent and identically distributed (i.i.d.) random variables, from a U(0,1) distribution, and “stop as soon as their union contains the interval (0,1).” We call such a union “a cover.” To find the distributions of N(δ), the stopping time random variable, we need the joint distribution of order statistics from a U(0,1) distribution. For each δ>0 and n=1,2,…, we obtain a general expression for P(N(δ)≤n), and for a fixed value of δ, it is the distribution function of N(δ). For a given n, let Δ(n) be the minimum value of δ, so that the union of the n δ− neighborhoods of the first n observations contains the interval (0,1). Because N(δ)≤n if and only if Δ(n)≤δ, the distributions of Δ(n) can be obtained by fixing n in the general expression for P(N(δ)≤n). To describe the impact of δ on the distribution of N(δ) and that of n on Δ(n), we sketch the graphs of distribution functions and the empirical distribution functions.Keywords: Neighborhoodstopping ruleuniform distributionSubject Classification: MSC 2020: 26B1562E15 ACKNOWLEDGMENTThanks to S. B. Patil and S. R. Rattihalli for their computational assistance.DISCLOSUREThe author has no conflicts of interest to report.
期刊介绍:
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