A (tight) upper bound for the length of confidence intervals with conditional coverage

TL;DR

This paper establishes a finite upper bound on the length of confidence intervals with conditional coverage when using certain selective inference procedures, demonstrating their superiority over sample-splitting methods.

Contribution

It provides the first finite upper bound for confidence interval length with conditional coverage under data carving and randomized response methods combined with the polyhedral approach.

Findings

Confidence intervals have bounded expected length under the studied procedures.

Data carving and randomized response methods outperform sample-splitting in interval length.

The results contrast with previous findings where polyhedral method alone yields infinite expected length.

Abstract

We show that two popular selective inference procedures, namely data carving (Fithian et al., 2017) and selection with a randomized response (Tian et al., 2018b), when combined with the polyhedral method (Lee et al., 2016), result in confidence intervals whose length is bounded. This contrasts results for confidence intervals based on the polyhedral method alone, whose expected length is typically infinite (Kivaranovic and Leeb, 2020). Moreover, we show that these two procedures always dominate corresponding sample-splitting methods in terms of interval length.