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

• URL: http://arxiv.org/abs/2505.11725v1
• Date: Fri, 16 May 2025 22:14:49 GMT
• Title: CLT and Edgeworth Expansion for m-out-of-n Bootstrap Estimators of The Studentized Median
• Authors: Imon Banerjee, Sayak Chakrabarty,
• Abstract summary: The m-out-of-n bootstrap approximates the distribution of a statistic by repeatedly drawing m subsamples without replacement from an original sample of size n.<n>Despite its broad applicability across econometrics, biostatistics, and machine learning, rigorous parameter-free guarantees for the soundness of the bootstrap have remained elusive.<n>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.
• Score: 4.174296652683762
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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