Confidence Intervals for High-Dimensional Linear Regression: Minimax Rates and Adaptivity

Loading...
Thumbnail Image
Penn collection
Statistics Papers
Degree type
Discipline
Subject
Adaptivity
confidence interval
coverage probability
expected length
high-dimensional linear regression
minimaxity
sparsity
Physical Sciences and Mathematics
Funder
Grant number
License
Copyright date
Distributor
Related resources
Author
Cai, Tony
Guo, Zijian
Contributor
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.

Advisor
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
2017-05-01
Journal title
The Annals of Statistics
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation
Collection