# Asymptotics of the Empirical Bootstrap Method Beyond Asymptotic Normality

arXiv: https://arxiv.org/abs/2011.11248 · HTML: https://mlsanity.com/paper/2011.11248/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2020.

## Built on by (most cited first, 3 of 3)

- [A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing](https://mlsanity.com/paper/2302.05686/) (2023)
- [Interventional Processes for Causal Uncertainty Quantification](https://mlsanity.com/paper/2410.14483/) (2024)
- [Centroid Approximation for Bootstrap: Improving Particle Quality at Inference](https://mlsanity.com/paper/2110.08720/) (2021)

## Abstract (arXiv)

One of the most commonly used methods for forming confidence intervals for statistical inference is the empirical bootstrap, which is especially expedient when the limiting distribution of the estimator is unknown. However, despite its ubiquitous role, its theoretical properties are still not well understood for non-asymptotically normal estimators. In this paper, under stability conditions, we establish the limiting distribution of the empirical bootstrap estimator, derive tight conditions for it to be asymptotically consistent, and quantify the speed of convergence. Moreover, we propose three alternative ways to use the bootstrap method to build confidence intervals with coverage guarantees. Finally, we illustrate the generality and tightness of our results by a series of examples, including uniform confidence bands, two-sample kernel tests, minmax stochastic programs and the empirical risk of stacked estimators.