A New Perspective on High Dimensional Confidence Intervals

TL;DR

This paper introduces a new approach to high-dimensional confidence intervals focusing on average coverage rather than individual coverage, using Relaxed Lasso to produce more representative intervals and analyzing bootstrap limitations.

Contribution

It proposes a novel perspective on confidence intervals emphasizing average coverage and introduces a Relaxed Lasso-based method for more accurate interval estimation.

Findings

Relaxed Lasso achieves approximately correct average coverage.

Debiased methods often do not contain original estimates.

Bootstrap remains inconsistent under the new coverage perspective.

Abstract

Classically, confidence intervals are required to have consistent coverage across all values of the parameter. However, this will inevitably break down if the underlying estimation procedure is biased. For this reason, many efforts have focused on debiased versions of the lasso for interval construction. In the process of debiasing, however, the connection to the original estimates are often obscured. In this work, we offer a different perspective focused on average coverage in contrast to individual coverage. This perspective results in confidence intervals that better reflect the original assumptions, as opposed to debiased intervals, which often do not even contain the original lasso estimates. To this end we propose a method based on the Relaxed Lasso that gives approximately correct average coverage and compare this to debiased methods which attempt to produce correct individual coverage. With this new definition of coverage we also briefly revisit the bootstrap, which Chatterjee and Lahiri (2010) showed was inconsistent for lasso, but find that it fails even under this alternative coverage definition.