HIGH-DIMENSIONAL INFLUENCE MEASURE

• 成果类型: Article
• 署名作者: Zhao, Junlong; Leng, Chenlei; Li, Lexin; Wang, Hansheng
• 署名单位: Beihang University; University of Warwick; National University of Singapore; North Carolina State University; Peking University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1165
• 发表日期: 2013
• 页码: 2639-2667
• 关键词: linear longitudinal models; influence diagnostics; adaptive lasso; mixed models; deletion diagnostics; REGRESSION SHRINKAGE; variable selection; oracle properties; cooks distance; perturbation

摘要

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.