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Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions
Fan, Jianqing1,2; Li, Quefeng3; Wang, Yuyan1,4
2017
Source PublicationJOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
ISSN1369-7412
Volume79Issue:1Pages:247-265
AbstractData subject to heavy-tailed errors are commonly encountered in various scientific fields. To address this problem, procedures based on quantile regression and least absolute deviation regression have been developed in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions, especially when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultrahigh dimensional settings with existence of only the second moment? To solve this problem, we propose a penalized Huber loss with diverging parameter to reduce biases created by the traditional Huber loss. Such a penalized robust approximate (RA) quadratic loss will be called the RA lasso. In the ultrahigh dimensional setting, where the dimensionality can grow exponentially with the sample size, our results reveal that the RA lasso estimator produces a consistent estimator at the same rate as the optimal rate under the light tail situation. We further study the computational convergence of the RA lasso and show that the composite gradient descent algorithm indeed produces a solution that admits the same optimal rate after sufficient iterations. As a by-product, we also establish the concentration inequality for estimating the population mean when there is only the second moment. We compare the RA lasso with other regularized robust estimators based on quantile regression and least absolute deviation regression. Extensive simulation studies demonstrate the satisfactory finite sample performance of the RA lasso.
KeywordHigh dimension Huber loss M-estimator Optimal rate Robust regularization
DOI10.1111/rssb.12166
Language英语
Funding ProjectNational Science Foundation[DMS-1206464] ; National Science Foundation[DMS-1406266] ; National Institutes of Health[R01-GM072611-9] ; National Institutes of Health[R01-GM100474-4]
WOS Research AreaMathematics
WOS SubjectStatistics & Probability
WOS IDWOS:000392486000012
PublisherWILEY-BLACKWELL
Citation statistics
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/24611
Collection中国科学院数学与系统科学研究院
Corresponding AuthorFan, Jianqing
Affiliation1.Princeton Univ, Princeton, NJ 08544 USA
2.Acad Math & Syst Sci, Beijing, Peoples R China
3.Univ N Carolina, Chapel Hill, NC USA
4.Princeton Univ, Princeton, NJ USA
Recommended Citation
GB/T 7714
Fan, Jianqing,Li, Quefeng,Wang, Yuyan. Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions[J]. JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY,2017,79(1):247-265.
APA Fan, Jianqing,Li, Quefeng,&Wang, Yuyan.(2017).Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions.JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY,79(1),247-265.
MLA Fan, Jianqing,et al."Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions".JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY 79.1(2017):247-265.
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