Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions

doi:10.1111/rssb.12166

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	Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions
	Fan, Jianqing 1,2; Li, Quefeng 3; Wang, Yuyan 1,4
	2017
发表期刊	JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
ISSN	1369-7412
卷号	79 期号:1 页码:247-265
摘要	Data 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.
关键词	High dimension Huber loss M-estimator Optimal rate Robust regularization
DOI	10.1111/rssb.12166
语种	英语
资助项目	National Science Foundation[DMS-1206464] ; National Science Foundation[DMS-1406266] ; National Institutes of Health[R01-GM072611-9] ; National Institutes of Health[R01-GM100474-4]
WOS研究方向	Mathematics
WOS类目	Statistics & Probability
WOS记录号	WOS:000392486000012
出版者	WILEY-BLACKWELL
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/24611
专题	中国科学院数学与系统科学研究院
通讯作者	Fan, Jianqing
作者单位	1.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
推荐引用方式 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.