CONFIDENCE SETS IN SPARSE REGRESSION

• 成果类型: Article
• 署名作者: Nickl, Richard; van de Geer, Sara
• 署名单位: University of Cambridge; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1170
• 发表日期: 2013
• 页码: 2852-2876
• 关键词: gaussian regression; DANTZIG SELECTOR; estimators; balls; bands

摘要

The problem of constructing confidence sets in the high-dimensional linear model with n response variables and p parameters, possibly p >= n, is considered. Full honest adaptive inference is possible if the rate of sparse estimation does not exceed n(-1/4), otherwise sparse adaptive confidence sets exist only over strict subsets of the parameter spaces for which sparse estimators exist. Necessary and sufficient conditions for the existence of confidence sets that adapt to a fixed sparsity level of the parameter vector are given in terms of minimal l(2)-separation conditions on the parameter space. The design conditions cover common coherence assumptions used in models for sparsity, including (possibly correlated) sub-Gaussian designs.