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April 2017 Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity
T. Tony Cai, Zijian Guo
Ann. Statist. 45(2): 615-646 (April 2017). DOI: 10.1214/16-AOS1461

Abstract

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.

Citation

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T. Tony Cai. Zijian Guo. "Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity." Ann. Statist. 45 (2) 615 - 646, April 2017. https://doi.org/10.1214/16-AOS1461

Information

Received: 1 June 2015; Revised: 1 February 2016; Published: April 2017
First available in Project Euclid: 16 May 2017

zbMATH: 1371.62045
MathSciNet: MR3650395
Digital Object Identifier: 10.1214/16-AOS1461

Subjects:
Primary: 62G15
Secondary: 62C20 , 62H35

Keywords: Adaptivity , Confidence interval , coverage probability , expected length , high-dimensional linear regression , minimaxity , Sparsity

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 2 • April 2017
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