随机集的新定义

Pub Date : 2023-06-30 DOI:10.3336/gm.58.1.10
Vesna Gotovac DJogaš, K. Helisova, L. Klebanov, J. Stanek, I. Volchenkova
{"title":"随机集的新定义","authors":"Vesna Gotovac DJogaš, K. Helisova, L. Klebanov, J. Stanek, I. Volchenkova","doi":"10.3336/gm.58.1.10","DOIUrl":null,"url":null,"abstract":"A new definition of random sets is proposed in the presented paper.\nIt is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets.\nMotivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases.\nWe place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations.\nA distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements.\nThe distance is then used to define a class of Borel subsets of B.A.\nConsequently, random sets are defined as measurable mappings taking values in the B.A.\nThis approach enables us to use more general family of distances between realisations of random sets\nwhich allows us to make new statistical tests concerning equality of some characteristics of random set distributions.\nAs an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A new definition of random set\",\"authors\":\"Vesna Gotovac DJogaš, K. Helisova, L. Klebanov, J. Stanek, I. Volchenkova\",\"doi\":\"10.3336/gm.58.1.10\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A new definition of random sets is proposed in the presented paper.\\nIt is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets.\\nMotivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases.\\nWe place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations.\\nA distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements.\\nThe distance is then used to define a class of Borel subsets of B.A.\\nConsequently, random sets are defined as measurable mappings taking values in the B.A.\\nThis approach enables us to use more general family of distances between realisations of random sets\\nwhich allows us to make new statistical tests concerning equality of some characteristics of random set distributions.\\nAs an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-06-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3336/gm.58.1.10\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3336/gm.58.1.10","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

本文提出了随机集的一个新定义。它基于可测空间中的特定距离,并使用负定核从初始空间延拓到随机集之一。引入新定义的动机是,经典方法处理随机集实现之间的豪斯多夫距离,这在许多情况下对统计分析是不满意的。我们将随机集的实现放在一个完全布尔代数(ba)中,赋予了一个正有限测度,旨在捕捉实现的重要特征。ba上的距离被引入为其两个元素之间对称差度量的平方根。因此,随机集被定义为在b.a中取值的可测量映射。这种方法使我们能够在随机集的实现之间使用更一般的距离,这使我们能够对随机集分布的某些特征的相等性进行新的统计检验。在此基础上,提出了新定义的随机集相对于交点的稳定性概念,并得到了极限定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
A new definition of random set
A new definition of random sets is proposed in the presented paper. It is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets. Motivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases. We place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations. A distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements. The distance is then used to define a class of Borel subsets of B.A. Consequently, random sets are defined as measurable mappings taking values in the B.A. This approach enables us to use more general family of distances between realisations of random sets which allows us to make new statistical tests concerning equality of some characteristics of random set distributions. As an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
×
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
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 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 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1