# CLT and Edgeworth Expansion for m-out-of-n Bootstrap Estimators of The Studentized Median

arXiv: https://arxiv.org/abs/2505.11725 · HTML: https://mlsanity.com/paper/2505.11725/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 75 references that are in the map.
- First posted on arXiv 16 May 2025.

## Builds on (most cited first, 13 of 13)

- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [mBLIP: Efficient Bootstrapping of Multilingual Vision-LLMs](https://mlsanity.com/paper/2307.06930/) (2023)
- [Partially Observed Markov Decision Processes. Problem Sets and Internet Supplement](https://mlsanity.com/paper/1604.08127/) (2016)
- [Online Bootstrap Inference For Policy Evaluation in Reinforcement Learning](https://mlsanity.com/paper/2108.03706/) (2021)
- [Efficient Toxic Content Detection by Bootstrapping and Distilling Large Language Models](https://mlsanity.com/paper/2312.08303/) (2023)
- [Free and Customizable Code Documentation with LLMs: A Fine-Tuning Approach](https://mlsanity.com/paper/2412.00726/) (2024)
- [On Applications of Bootstrap in Continuous Space Reinforcement Learning](https://mlsanity.com/paper/1903.05803/) (2019)
- [Single-Pass Pivot Algorithm for Correlation Clustering. Keep it simple!](https://mlsanity.com/paper/2305.13560/) (2023)
- [Offline Estimation of Controlled Markov Chains: Minimaxity and Sample Complexity](https://mlsanity.com/paper/2211.07092/) (2022)
- [On the Consistency of Maximum Likelihood Estimation of Probabilistic Principal Component Analysis](https://mlsanity.com/paper/2311.05046/) (2023)
- [MM-PoE: Multiple Choice Reasoning via. Process of Elimination using Multi-Modal Models](https://mlsanity.com/paper/2412.07148/) (2024)
- [Bootstrap Model Aggregation for Distributed Statistical Learning](https://mlsanity.com/paper/1607.01036/) (2016)
- [Bootstrap Your Own Context Length](https://mlsanity.com/paper/2412.18860/) (2024)

## Built on by (most cited first, 2 of 2)

- [Time-Constrained Recommendations: Reinforcement Learning Strategies for E-Commerce](https://mlsanity.com/paper/2512.13726/) (2025)
- [PixRec: Leveraging Visual Context for Next-Item Prediction in Sequential Recommendation](https://mlsanity.com/paper/2601.06458/) (2026)

## Abstract (arXiv)

The m-out-of-n bootstrap, originally proposed by Bickel, Gotze, and Zwet (1992), approximates the distribution of a statistic by repeatedly drawing m subsamples (with m much smaller than n) without replacement from an original sample of size n. It is now routinely used for robust inference with heavy-tailed data, bandwidth selection, and other large-sample applications. Despite its broad applicability across econometrics, biostatistics, and machine learning, rigorous parameter-free guarantees for the soundness of the m-out-of-n bootstrap when estimating sample quantiles have remained elusive. This paper establishes such guarantees by analyzing the estimator of sample quantiles obtained from m-out-of-n resampling of a dataset of size n. We first prove a central limit theorem for a fully data-driven version of the estimator that holds under a mild moment condition and involves no unknown nuisance parameters. We then show that the moment assumption is essentially tight by constructing a counter-example in which the CLT fails. Strengthening the assumptions slightly, we derive an Edgeworth expansion that provides exact convergence rates and, as a corollary, a Berry Esseen bound on the bootstrap approximation error. Finally, we illustrate the scope of our results by deriving parameter-free asymptotic distributions for practical statistics, including the quantiles for random walk Metropolis-Hastings and the rewards of ergodic Markov decision processes, thereby demonstrating the usefulness of our theory in modern estimation and learning tasks.