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

Imon Banerjee, Sayak Chakrabarty

Abstract

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.

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

Abstract

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.
Paper Structure (55 sections, 27 theorems, 242 equations, 3 tables)

This paper contains 55 sections, 27 theorems, 242 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Background and Related Research
  3. Problem Statement
  4. Data-generating process.
  5. Studentization.
  6. Studentization.
  7. Goals
  8. Theoretical Results
  9. Sketch of proof:
  10. Importance of the Moment Condition
  11. Convergence Rates Via Edgeworth Expansions
  12. Sketch of Proof:
  13. On the choice of $m$:
  14. Extensions to Regenerating Markov Chains
  15. Sketch of Proof:
  16. ...and 40 more sections

Key Result

Theorem 1

Let $\mu$ be the unique $p$-th quantile and $F\in\mathbb{S}_1(\mu)$. Furthermore, let $E|X_1|^\alpha < \infty$ for some $\alpha > 0$. Then for any $m=\mathdutchcal{o}(n)$, and $m,n\rightarrow\infty$. Furthermore,

Theorems & Definitions (57)

  • Remark
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Remark
  • Corollary 1
  • Definition 1
  • Remark
  • Remark
  • Theorem 3
  • ...and 47 more