Non-asymptotic robustness analysis of regression depth median

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Yijun Zuo
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引用次数: 1

Abstract

The maximum depth estimator (aka depth median) (βRD) 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, βRD 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 ϵ-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 βRD were obtained whereas an exact FSBP for βRD remained open in the last twenty-plus years. Furthermore, is the asymptotic breakdown value 1/3 (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 βRD and the newly obtained exact FSBP. It justifies the employment of βRD as a robust alternative to the traditional estimators and demonstrates the necessity and the merit of using the FSBP in finite-sample real practice.

回归深度中位数的非渐近稳健性分析
由Rousseeuw和Hubert(1999)的回归深度(RD)导出的最大深度估计量(又称深度中位数)(βRD *)是回归中最流行的估计量之一。它具有与单变量定位对应物相似的出色鲁棒性。事实上,βRD *可以渐近地抵抗高达33%的污染而不破裂,而传统的(最小二乘和最小绝对偏差)估计器则为0%(见Van Aelst和Rousseeuw(2000))。Van Aelst和Rousseeuw(2000)的结果是开创性的,但它们仅限于回归对称种群(具有严格的正密度),ϵ-contamination,最大偏差模型和渐近意义。对于固定的有限样本大小的实践,对于估计器来说,最普遍的鲁棒性度量是有限样本击穿点(FSBP) (Donoho和Huber, 1983)。尽管在文献中做了许多尝试,但在βRD *的FSBP上只获得了零星的部分结果,而βRD *的确切FSBP在过去的20多年里仍然是开放的。此外,在有限样本实践中,渐近击穿值1/3(有限样本击穿值递增序列的极限)是否相关?(或者有限样本和极限击穿值之间有什么区别?)这样的讨论还没有在文献中给出。本文解决了上述问题,揭示了βRD *的回归深度与新获得的精确FSBP之间的内在联系。它证明了βRD *作为传统估计器的鲁棒替代,并证明了在有限样本实际实践中使用FSBP的必要性和优点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
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.
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