2025/05/16 by Imon Banerjee, Banerjee, Imon, Sayak Chakrabarty +1 · 1 citation
Computer Science · Mathematics · #Artificial Intelligence (cs.AI) #Computational Engineering #FOS: Computer and information sciences #FOS: Mathematics #Finance #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods #Methodology (stat.ME) #Statistical Methods and Inference #Statistics Theory (math.ST) #and Science (cs.CE)
paper · pdf · doi:10.48550/arxiv.2505.11725
openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.