Group bound: confidence intervals for groups of variables in sparse high dimensional regression without assumptions on the design

• 成果类型: Article
• 署名作者: Meinshausen, Nicolai
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/rssb.12094
• 发表日期: 2015
• 页码: 923-945
• 关键词: lasso; selection

摘要

It is in general challenging to provide confidence intervals for individual variables in high dimensional regression without making strict or unverifiable assumptions on the design matrix. We show here that a 'group bound' confidence interval can be derived without making any assumptions on the design matrix. The lower bound for the regression coefficient of individual variables can be derived via linear programming. The idea also generalizes naturally to groups of variables, where we can derive a one-sided confidence interval for the joint effect of a group. Although the confidence intervals of individual variables are by the nature of the problem often very wide, it is shown to be possible to detect the contribution of groups of highly correlated predictor variables even when no variable individually shows a significant effect. The assumptions that are necessary to detect the effect of groups of variables are shown to be weaker than the weakest known assumptions that are necessary to detect the effect of individual variables.