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

TL;DR

This paper introduces a general, asymptotically optimal method for constructing confidence regions and tests for low-dimensional components in high-dimensional models, applicable to linear and generalized linear models with various design types.

Contribution

It extends existing methods by analyzing their asymptotic properties and establishing optimality, while also adapting to generalized linear models and correlated designs.

Findings

Method achieves asymptotic optimality in linear models.

Extends to generalized linear models with convex loss functions.

Provides theoretical analysis for Gaussian, sub-Gaussian, and bounded designs.

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.