{"title":"用平衡不完全U-统计量逼近散射的对称估计","authors":"Lutz Dümbgen, Klaus Nordhausen","doi":"10.1007/s10463-023-00879-1","DOIUrl":null,"url":null,"abstract":"<div><p>We derive limiting distributions of symmetrized estimators of scatter. Instead of considering all <span>\\(n(n-1)/2\\)</span> pairs of the <i>n</i> observations, we only use <i>nd</i> suitably chosen pairs, where <span>\\(d \\ge 1\\)</span> is substantially smaller than <i>n</i>. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever <span>\\(d = d(n) \\rightarrow \\infty\\)</span> at arbitrarily slow speed. We also investigate the asymptotic properties for arbitrary fixed <i>d</i>. These considerations and numerical examples indicate that for practical purposes, moderate fixed values of <i>d</i> between 10 and 20 yield already estimators which are computationally feasible and rather close to the original ones.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Approximating symmetrized estimators of scatter via balanced incomplete U-statistics\",\"authors\":\"Lutz Dümbgen, Klaus Nordhausen\",\"doi\":\"10.1007/s10463-023-00879-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We derive limiting distributions of symmetrized estimators of scatter. Instead of considering all <span>\\\\(n(n-1)/2\\\\)</span> pairs of the <i>n</i> observations, we only use <i>nd</i> suitably chosen pairs, where <span>\\\\(d \\\\ge 1\\\\)</span> is substantially smaller than <i>n</i>. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever <span>\\\\(d = d(n) \\\\rightarrow \\\\infty\\\\)</span> at arbitrarily slow speed. We also investigate the asymptotic properties for arbitrary fixed <i>d</i>. These considerations and numerical examples indicate that for practical purposes, moderate fixed values of <i>d</i> between 10 and 20 yield already estimators which are computationally feasible and rather close to the original ones.</p></div>\",\"PeriodicalId\":55511,\"journal\":{\"name\":\"Annals of the Institute of Statistical Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-08-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of the Institute of Statistical Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10463-023-00879-1\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of the Institute of Statistical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10463-023-00879-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
摘要
我们推导出散点对称估计值的极限分布。我们不考虑 n 个观测值中的所有 (n(n-1)/2\)对,而只使用 nd 个适当选择的对,其中 (d\ge 1\)大大小于 n。事实证明,当 (d = d(n) \rightarrow \infty\)以任意慢的速度时,所得到的估计值在渐近上等同于原始估计值。我们还研究了任意固定 d 的渐近特性。这些考虑因素和数值示例表明,在实际应用中,介于 10 到 20 之间的适度固定 d 值所得到的估计值在计算上是可行的,而且与原始估计值相当接近。
Approximating symmetrized estimators of scatter via balanced incomplete U-statistics
We derive limiting distributions of symmetrized estimators of scatter. Instead of considering all \(n(n-1)/2\) pairs of the n observations, we only use nd suitably chosen pairs, where \(d \ge 1\) is substantially smaller than n. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever \(d = d(n) \rightarrow \infty\) at arbitrarily slow speed. We also investigate the asymptotic properties for arbitrary fixed d. These considerations and numerical examples indicate that for practical purposes, moderate fixed values of d between 10 and 20 yield already estimators which are computationally feasible and rather close to the original ones.
期刊介绍:
Annals of the Institute of Statistical Mathematics (AISM) aims to provide a forum for open communication among statisticians, and to contribute to the advancement of statistics as a science to enable humans to handle information in order to cope with uncertainties. It publishes high-quality papers that shed new light on the theoretical, computational and/or methodological aspects of statistical science. Emphasis is placed on (a) development of new methodologies motivated by real data, (b) development of unifying theories, and (c) analysis and improvement of existing methodologies and theories.