{"title":"尾巴从哪里开始?作为边界的拐点和最大曲率","authors":"Rafael Cabral, Maria de Iorio, Andrea Cremaschi","doi":"arxiv-2409.06308","DOIUrl":null,"url":null,"abstract":"Understanding the tail behavior of distributions is crucial in statistical\ntheory. For instance, the tail of a distribution plays a ubiquitous role in\nextreme value statistics, where it is associated with the likelihood of extreme\nevents. There are several ways to characterize the tail of a distribution based\non how the tail function, $\\bar{F}(x) = P(X>x)$, behaves when $x\\to\\infty$.\nHowever, for unimodal distributions, where does the core of the distribution\nend and the tail begin? This paper addresses this unresolved question and\nexplores the usage of delimiting points obtained from the derivatives of the\ndensity function of continuous random variables, namely, the inflection point\nand the point of maximum curvature. These points are used to delimit the bulk\nof the distribution from its tails. We discuss the estimation of these\ndelimiting points and compare them with other measures associated with the tail\nof a distribution, such as the kurtosis and extreme quantiles. We derive the\nproposed delimiting points for several known distributions and show that it can\nbe a reasonable criterion for defining the starting point of the tail of a\ndistribution.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"58 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Where does the tail start? Inflection Points and Maximum Curvature as Boundaries\",\"authors\":\"Rafael Cabral, Maria de Iorio, Andrea Cremaschi\",\"doi\":\"arxiv-2409.06308\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Understanding the tail behavior of distributions is crucial in statistical\\ntheory. For instance, the tail of a distribution plays a ubiquitous role in\\nextreme value statistics, where it is associated with the likelihood of extreme\\nevents. There are several ways to characterize the tail of a distribution based\\non how the tail function, $\\\\bar{F}(x) = P(X>x)$, behaves when $x\\\\to\\\\infty$.\\nHowever, for unimodal distributions, where does the core of the distribution\\nend and the tail begin? This paper addresses this unresolved question and\\nexplores the usage of delimiting points obtained from the derivatives of the\\ndensity function of continuous random variables, namely, the inflection point\\nand the point of maximum curvature. These points are used to delimit the bulk\\nof the distribution from its tails. We discuss the estimation of these\\ndelimiting points and compare them with other measures associated with the tail\\nof a distribution, such as the kurtosis and extreme quantiles. We derive the\\nproposed delimiting points for several known distributions and show that it can\\nbe a reasonable criterion for defining the starting point of the tail of a\\ndistribution.\",\"PeriodicalId\":501379,\"journal\":{\"name\":\"arXiv - STAT - Statistics Theory\",\"volume\":\"58 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-10\",\"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-2409.06308\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - STAT - Statistics Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06308","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Where does the tail start? Inflection Points and Maximum Curvature as Boundaries
Understanding the tail behavior of distributions is crucial in statistical
theory. For instance, the tail of a distribution plays a ubiquitous role in
extreme value statistics, where it is associated with the likelihood of extreme
events. There are several ways to characterize the tail of a distribution based
on how the tail function, $\bar{F}(x) = P(X>x)$, behaves when $x\to\infty$.
However, for unimodal distributions, where does the core of the distribution
end and the tail begin? This paper addresses this unresolved question and
explores the usage of delimiting points obtained from the derivatives of the
density function of continuous random variables, namely, the inflection point
and the point of maximum curvature. These points are used to delimit the bulk
of the distribution from its tails. We discuss the estimation of these
delimiting points and compare them with other measures associated with the tail
of a distribution, such as the kurtosis and extreme quantiles. We derive the
proposed delimiting points for several known distributions and show that it can
be a reasonable criterion for defining the starting point of the tail of a
distribution.