TESTABILITY OF HIGH-DIMENSIONAL LINEAR MODELS WITH NONSPARSE STRUCTURES

Jelena Bradic, Jianqing Fan, Yinchu Zhu

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Understanding statistical inference under possibly nonsparse highdimensional 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 √ n rate, regardless of the size of the model sparsity.

Original languageEnglish (US)
Pages (from-to)615-639
Number of pages25
JournalAnnals of Statistics
Volume50
Issue number2
DOIs
StatePublished - Apr 2022

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • confidence intervals
  • Minimax theory
  • uniform nontestability
  • ℓ2-constraint

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