High-dimensional Edgeworth expansion for the empirical bootstrap

Establish that the empirical bootstrap satisfies the high-dimensional Edgeworth expansion condition (formal-ae-boot) with a vanishing and explicit error term Δ_n under suitable assumptions, thereby extending the coverage error bounds in Theorem 2.1 beyond the wild bootstrap.

Background

The coverage error bounds in Theorem 2.1 are derived under an Edgeworth expansion assumption for the bootstrap law (equation (formal-ae-boot)). The paper verifies this condition for certain wild bootstraps but not for the empirical bootstrap in high dimensions.

The authors explicitly note the absence of results establishing (formal-ae-boot) with a reasonable Δ_n for the empirical bootstrap, leaving a gap in theory for one of the most commonly used resampling methods.