On asymptotically optimal confidence regions and tests for high-dimensional models

Sara Van De Geer, Peter Bühlmann, Ya'acov Ritov, Ruben Dezeure

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671 Scopus citations

Abstract

We propose a general method for constructing confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in a high-dimensional model. It can be easily adjusted for multiplicity taking dependence among tests into account. For linear models, our method is essentially the same as in Zhang and Zhang [J. R. Stat. Soc. Ser. B Stat. Methodol. 76 (2014) 217-242]: we analyze its asymptotic properties and establish its asymptotic optimality in terms of semiparametric efficiency. Our method naturally extends to generalized linear models with convex loss functions. We develop the corresponding theory which includes a careful analysis for Gaussian, sub-Gaussian and bounded correlated designs.

Original languageEnglish
Pages (from-to)1166-1202
Number of pages37
JournalAnnals of Statistics
Volume42
Issue number3
DOIs
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© Institute of Mathematical Statistics, 2014.

Keywords

  • Central limit theorem
  • Generalized linear model
  • Lasso
  • Linear model
  • Multiple testing
  • Semiparametric efficiency
  • Sparsity

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