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

TL;DR

This paper rigorously analyzes the m-out-of-n bootstrap for sample quantiles, establishing CLTs, Edgeworth expansions, and Berry-Esseen bounds, with applications to robust inference in complex stochastic models.

Contribution

It provides the first parameter-free CLT and Edgeworth expansion for m-out-of-n bootstrap estimators of the median, with tight assumptions and practical applications.

Findings

Proves a data-driven CLT for bootstrap quantile estimators.

Derives an Edgeworth expansion with explicit convergence rates.

Demonstrates applicability to Markov chain-based statistics.

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.