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

2025-05-16

AI Summary

This paper addresses the lack of distribution-free theoretical guarantees for the *m*-out-of-*n* bootstrap in quantile estimation—particularly for studentized medians. Under mild moment conditions, it establishes, for the first time, a central limit theorem and an Edgeworth expansion for this method, without tuning parameters or assuming a specific underlying distribution. Methodologically, the analysis integrates sampling-without-replacement, studentization, Berry–Esseen bounds, and stochastic process techniques to derive precise convergence rates and explicit error bounds. Key contributions include: (1) the first distribution-free theoretical framework for bootstrap-based quantile inference; (2) proof of the tightness of the required moment conditions; (3) extension to modern learning settings—including MCMC (e.g., quantile inference for random-walk Metropolis–Hastings) and ergodic Markov decision processes (e.g., reward distribution estimation); and (4) theoretical validity under heavy-tailed data and robust statistical inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

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.

Problem

• Establishes guarantees for m-out-of-n bootstrap in quantile estimation
• Proves central limit theorem for data-driven quantile estimators
• Derives Edgeworth expansion for exact bootstrap convergence rates

Innovation

• Uses m-out-of-n bootstrap for robust inference
• Proves central limit theorem for data-driven estimator
• Derives Edgeworth expansion for exact convergence rates