Studentized Cheap Bootstrap: Achieving Higher-Order Coverage Accuracy with Low Computation

2026-06-24

AI Summary

This work proposes a “cheap studentized bootstrap” that achieves the same high-order coverage accuracy as the conventional studentized bootstrap while drastically reducing computational cost. Traditional studentized bootstrap methods require extensive resampling or analytical standard error calculations, rendering them computationally expensive. The key innovation lies in formally establishing, for the first time, the connection between studentized statistics and the t-distribution, revealing that the degrees of freedom in the t-distribution reflect the amount of resampling computation rather than the original sample size. Building on Edgeworth and Cornish–Fisher expansions together with limiting t-distribution theory, the authors construct a high-order accuracy framework that maintains rigorous theoretical guarantees with only a minimal number of Monte Carlo resamples.

Abstract

The bootstrap is a versatile method for quantifying statistical uncertainty. Among its variants, a popular approach, the studentized bootstrap, provably achieves higher-order coverage error reduction compared to other benchmarks. However, its implementation typically requires an analytical form of the standard error, or otherwise an additional layer of resampling effort which can be computationally expensive. In this paper, we introduce what we call the studentized cheap bootstrap that achieves the same higher-order coverage accuracy as the conventional studentization, but substantially thinning the computational effort in the additional resampling layer to only very few Monte Carlo replications. Intriguingly, while conventional wisdom views "studentization" as an informal link between the bootstrap and t-distribution, we provide a first recognition that this link is in fact formal, notably with a distinct insight that the degree of freedom in the t-distribution corresponds to the Monte Carlo computation effort in the additional resampling layer, rather than the data size as in traditional thinking. Moreover, our desirable higher-order coverage accuracy builds crucially on this insight, as well as explicit calculations and geometric analyses of higher-order terms in the Edgeworth and Cornish-Fisher expansions tailored to limiting t-distributions.

Problem

• studentized bootstrap
• coverage accuracy
• computational efficiency
• higher-order approximation
• Monte Carlo resampling

Innovation

• studentized bootstrap
• cheap bootstrap
• higher-order coverage accuracy
• t-distribution
• Monte Carlo resampling