The Annals of Statistics

Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity

T. Tony Cai and Zijian Guo

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Confidence sets play a fundamental role in statistical inference. In this paper, we consider confidence intervals for high-dimensional linear regression with random design. We first establish the convergence rates of the minimax expected length for confidence intervals in the oracle setting where the sparsity parameter is given. The focus is then on the problem of adaptation to sparsity for the construction of confidence intervals. Ideally, an adaptive confidence interval should have its length automatically adjusted to the sparsity of the unknown regression vector, while maintaining a pre-specified coverage probability. It is shown that such a goal is in general not attainable, except when the sparsity parameter is restricted to a small region over which the confidence intervals have the optimal length of the usual parametric rate. It is further demonstrated that the lack of adaptivity is not due to the conservativeness of the minimax framework, but is fundamentally caused by the difficulty of learning the bias accurately.

Article information

Ann. Statist., Volume 45, Number 2 (2017), 615-646.

Received: June 2015
Revised: February 2016
First available in Project Euclid: 16 May 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G15: Tolerance and confidence regions
Secondary: 62C20: Minimax procedures 62H35: Image analysis

Adaptivity confidence interval coverage probability expected length high-dimensional linear regression minimaxity sparsity


Cai, T. Tony; Guo, Zijian. Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity. Ann. Statist. 45 (2017), no. 2, 615--646. doi:10.1214/16-AOS1461.

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Supplemental materials

  • Supplement to “Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity”. Detailed proofs of the adaptivity lower bound and minimax upper bound for confidence intervals of the linear functional $\xi^{\intercal}\beta$ with a dense loading $\xi$ are given. The minimax rates and adaptivity of confidence intervals of the linear functional $\xi^{\intercal}\beta$ are established when there is prior knowledge that $\Omega=\mathrm{I}$ and $\sigma=\sigma_{0}$. Extra propositions and technical lemmas are also proved in the supplement.