Zhao, Junlong, Leng, Chenlei, Li, Lexin and Wang, Hansheng (2013) High-dimensional influence measure. The Annals of Statistics, Volume 41 (Number 5). pp. 2639-2667. doi:10.1214/13-AOS1165

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Abstract

Influence diagnosis is important since presence of influential observations could lead to distorted analysis and misleading interpretations. For high dimensional data, it is particularly so, as the increased dimensionality and complexity may amplify both the chance of an observation being influential, and its potential impact on the analysis. In this article, we propose a novel high dimensional influence measure for regressions with the number of predictors far exceeding the sample size. Our proposal can be viewed as a high dimensional counterpart to the classical Cook's distance. However, whereas the Cook's distance quantifies the individual observation's influence on the least squares regression coefficient estimate, our new diagnosis measure captures the influence on the marginal correlations, which in turn exerts serious influence on downstream analysis including coefficient estimation, variable selection and screening. Moreover, we establish the asymptotic distribution of the proposed influence measure by letting the predictor dimension go to infinity. Availability of this asymptotic distribution leads to a principled rule to determine the critical value for influential observation detection. Both simulations and real data analysis demonstrate usefulness of the new influence diagnosis measure.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Mathematical statistics, Asymptotic distribution (Probability theory)
Journal or Publication Title:	The Annals of Statistics
Publisher:	Institute of Mathematical Statistics
ISSN:	0090-5364
Official Date:	2013
Dates:	Date Event 2013 Published
Volume:	Volume 41
Number:	Number 5
Page Range:	pp. 2639-2667
DOI:	10.1214/13-AOS1165
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	National Science Foundation (U.S.) (NSF), Guo jia zi ran ke xue ji jin wei yuan hui (China) [National Natural Science Foundation of China] (NSFC), National University of Singapore, Fox Ying Tong Education Foundation, Beijing da xue [Peking University], China. Jiao yu bu [Ministry of Education]
Grant number:	DMS-11-06668 (NSF); 11101022, 11131002, 11271032 (NSFC);

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