SOME NONASYMPTOTIC RESULTS ON RESAMPLING IN HIGH DIMENSION, I: CONFIDENCE REGIONS

• 成果类型: Article
• 署名作者: Arlot, Sylvain; Blanchard, Gilles; Roquain, Etienne
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - Institute for Information Sciences & Technologies (INS2I); Inria; Universite PSL; Ecole Normale Superieure (ENS); Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/08-AOS667
• 发表日期: 2010
• 页码: 51-82
• 关键词: Bootstrap; localization; algorithms

摘要

We Study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a nonasymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second oil a resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of interest in their own right. We also discuss the question of accuracy when using Monte Carlo approximations of the resampled quantities.