{"title":"回归深度中位数的非渐近稳健性分析","authors":"Yijun Zuo","doi":"10.1016/j.jmva.2023.105247","DOIUrl":null,"url":null,"abstract":"<div><p>The maximum depth estimator (aka depth median) (<span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span>) induced from regression depth (RD) of Rousseeuw and Hubert (1999) is one of the most prevailing estimators in regression. It possesses outstanding robustness similar to the univariate location counterpart. Indeed, <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> can, asymptotically, resist up to 33% contamination without breakdown, in contrast to the 0% for the traditional (least squares and least absolute deviations) estimators (see Van Aelst and Rousseeuw (2000)). The results from Van Aelst and Rousseeuw (2000) are pioneering, yet they are limited to regression-symmetric populations (with a strictly positive density), the <span><math><mi>ϵ</mi></math></span>-contamination, maximum-bias model, and in asymptotical sense. With a fixed finite-sample size practice, the most prevailing measure of robustness for estimators is the finite-sample breakdown point (FSBP) (Donoho and Huber, 1983). Despite many attempts made in the literature, only sporadic partial results on FSBP for <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> were obtained whereas an exact FSBP for <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> remained open in the last twenty-plus years. Furthermore, is the asymptotic breakdown value <span><math><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></math></span> (the limit of an increasing sequence of finite-sample breakdown values) relevant in the finite-sample practice? (Or what is the difference between the finite-sample and the limit breakdown values?). Such discussions are yet to be given in the literature. This article addresses the above issues, revealing an intrinsic connection between the regression depth of <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> and the newly obtained exact FSBP. It justifies the employment of <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> as a robust alternative to the traditional estimators and demonstrates the necessity and the merit of using the FSBP in finite-sample real practice.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0047259X23000933/pdfft?md5=41b0163d4b47acc16c5399dda63160ea&pid=1-s2.0-S0047259X23000933-main.pdf","citationCount":"1","resultStr":"{\"title\":\"Non-asymptotic robustness analysis of regression depth median\",\"authors\":\"Yijun Zuo\",\"doi\":\"10.1016/j.jmva.2023.105247\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The maximum depth estimator (aka depth median) (<span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span>) induced from regression depth (RD) of Rousseeuw and Hubert (1999) is one of the most prevailing estimators in regression. It possesses outstanding robustness similar to the univariate location counterpart. Indeed, <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> can, asymptotically, resist up to 33% contamination without breakdown, in contrast to the 0% for the traditional (least squares and least absolute deviations) estimators (see Van Aelst and Rousseeuw (2000)). The results from Van Aelst and Rousseeuw (2000) are pioneering, yet they are limited to regression-symmetric populations (with a strictly positive density), the <span><math><mi>ϵ</mi></math></span>-contamination, maximum-bias model, and in asymptotical sense. With a fixed finite-sample size practice, the most prevailing measure of robustness for estimators is the finite-sample breakdown point (FSBP) (Donoho and Huber, 1983). Despite many attempts made in the literature, only sporadic partial results on FSBP for <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> were obtained whereas an exact FSBP for <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> remained open in the last twenty-plus years. Furthermore, is the asymptotic breakdown value <span><math><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></math></span> (the limit of an increasing sequence of finite-sample breakdown values) relevant in the finite-sample practice? (Or what is the difference between the finite-sample and the limit breakdown values?). Such discussions are yet to be given in the literature. This article addresses the above issues, revealing an intrinsic connection between the regression depth of <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> and the newly obtained exact FSBP. It justifies the employment of <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>R</mi><mi>D</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> as a robust alternative to the traditional estimators and demonstrates the necessity and the merit of using the FSBP in finite-sample real practice.</p></div>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2023-11-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0047259X23000933/pdfft?md5=41b0163d4b47acc16c5399dda63160ea&pid=1-s2.0-S0047259X23000933-main.pdf\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0047259X23000933\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0047259X23000933","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Non-asymptotic robustness analysis of regression depth median
The maximum depth estimator (aka depth median) () induced from regression depth (RD) of Rousseeuw and Hubert (1999) is one of the most prevailing estimators in regression. It possesses outstanding robustness similar to the univariate location counterpart. Indeed, can, asymptotically, resist up to 33% contamination without breakdown, in contrast to the 0% for the traditional (least squares and least absolute deviations) estimators (see Van Aelst and Rousseeuw (2000)). The results from Van Aelst and Rousseeuw (2000) are pioneering, yet they are limited to regression-symmetric populations (with a strictly positive density), the -contamination, maximum-bias model, and in asymptotical sense. With a fixed finite-sample size practice, the most prevailing measure of robustness for estimators is the finite-sample breakdown point (FSBP) (Donoho and Huber, 1983). Despite many attempts made in the literature, only sporadic partial results on FSBP for were obtained whereas an exact FSBP for remained open in the last twenty-plus years. Furthermore, is the asymptotic breakdown value (the limit of an increasing sequence of finite-sample breakdown values) relevant in the finite-sample practice? (Or what is the difference between the finite-sample and the limit breakdown values?). Such discussions are yet to be given in the literature. This article addresses the above issues, revealing an intrinsic connection between the regression depth of and the newly obtained exact FSBP. It justifies the employment of as a robust alternative to the traditional estimators and demonstrates the necessity and the merit of using the FSBP in finite-sample real practice.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.