TESTABILITY OF HIGH-DIMENSIONAL LINEAR MODELS WITH NONSPARSE STRUCTURES

• 成果类型: Article
• 署名作者: Bradic, Jelena; Fan, Jianqing; Zhu, Yinchu
• 署名单位: University of California System; University of California San Diego; University of California System; University of California San Diego; Princeton University; Brandeis University; Brandeis University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1932
• 发表日期: 2022
• 页码: 615-639
• 关键词: confidence-intervals; Minimax Rates; inference; regression; selection; Lasso

摘要

Understanding statistical inference under possibly nonsparse high-dimensional models has gained much interest recently. For a given component of the regression coefficient, we show that the difficulty of the problem depends on the sparsity of the corresponding row of the precision matrix of the covariates, not the sparsity of the regression coefficients. We develop new concepts of uniform and essentially uniform nontestability that allow the study of limitations of tests across a broad set of alternatives. Uniform nontestability identifies a collection of alternatives such that the power of any test, against any alternative in the group, is asymptotically at most equal to the nominal size. Implications of the new constructions include new minimax testability results that, in sharp contrast to the current results, do not depend on the sparsity of the regression parameters. We identify new tradeoffs between testability and feature correlation. In particular, we show that, in models with weak feature correlations, minimax lower bound can be attained by a test whose power has the root n rate, regardless of the size of the model sparsity.