Asymptotic coverage probabilities of bootstrap percentile confidence intervals for constrained parameters

TL;DR

This paper analyzes the asymptotic coverage probabilities of bootstrap percentile confidence intervals for parameters constrained by linear inequalities, especially near boundaries, revealing potential over- or under-coverage.

Contribution

It introduces a local asymptotic framework to understand coverage behavior near boundaries and provides theoretical insights for constrained bootstrap inference.

Findings

Coverage probabilities can exceed nominal levels near boundaries.

One-sample intervals tend to over-cover near constraints.

Two-sample intervals may under- or over-cover depending on the situation.

Abstract

The asymptotic behaviour of the commonly used bootstrap percentile confidence interval is investigated when the parameters are subject to linear inequality constraints. We concentrate on the important one- and two-sample problems with data generated from general parametric distributions in the natural exponential family. The focus of this paper is on quantifying the coverage probabilities of the parametric bootstrap percentile confidence intervals, in particular their limiting behaviour near boundaries. We propose a local asymptotic framework to study this subtle coverage behaviour. Under this framework, we discover that when the true parameters are on, or close to, the restriction boundary, the asymptotic coverage probabilities can always exceed the nominal level in the one-sample case; however, they can be, remarkably, both under and over the nominal level in the two-sample case. Using illustrative examples, we show that the results provide theoretical justification and guidance on applying the bootstrap percentile method to constrained inference problems.