# Exact adaptive confidence intervals for linear regression coefficients

## TL;DR

This paper introduces an adaptive confidence interval method for linear regression coefficients that guarantees exact frequentist coverage while minimizing average interval width, combining Bayesian and frequentist principles.

## Contribution

It develops a novel adaptive confidence interval procedure with exact coverage that is both data-driven and asymptotically Bayes-optimal in high-dimensional settings.

## Key findings

- Provides smaller intervals than traditional methods on average.
- Maintains exact $1-eta$ coverage non-asymptotically.
- Achieves asymptotic Bayes-optimality as p grows with n.

## Abstract

We propose an adaptive confidence interval procedure (CIP) for the coefficients in the normal linear regression model. This procedure has a frequentist coverage rate that is constant as a function of the model parameters, yet provides smaller intervals than the usual interval procedure, on average across regression coefficients. The proposed procedure is obtained by defining a class of CIPs that all have exact $1-\alpha$ frequentist coverage, and then selecting from this class the procedure that minimizes a prior expected interval width. Such a procedure may be described as "frequentist, assisted by Bayes" or FAB. We describe an adaptive approach for estimating the prior distribution from the data so that exact non-asymptotic $1-\alpha$ coverage is maintained. Additionally, in a "$p$ growing with $n$" asymptotic scenario, this adaptive FAB procedure is asymptotically Bayes-optimal among $1-\alpha$ frequentist CIPs.