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December 2022 On optimal block resampling for Gaussian-subordinated long-range dependent processes
Qihao Zhang, Soumendra N. Lahiri, Daniel J. Nordman
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Ann. Statist. 50(6): 3619-3646 (December 2022). DOI: 10.1214/22-AOS2242


Block-based resampling estimators have been intensively investigated for weakly dependent time processes, which has helped to inform implementation (e.g., best block sizes). However, little is known about resampling performance and block sizes under strong or long-range dependence. To establish guideposts in block selection, we consider a broad class of strongly dependent time processes, formed by a transformation of a stationary long-memory Gaussian series, and examine block-based resampling estimators for the variance of the prototypical sample mean; extensions to general statistical functionals are also considered. Unlike weak dependence, the properties of resampling estimators under strong dependence are shown to depend intricately on the nature of nonlinearity in the time series (beyond Hermite ranks) in addition to the long-memory coefficient and block size. Additionally, the intuition has often been that optimal block sizes should be larger under strong dependence (say O(n1/2) for a sample size n) than the optimal order O(n1/3) known under weak dependence. This intuition turns out to be largely incorrect, though a block order O(n1/2) may be reasonable (and even optimal) in many cases, owing to nonlinearity in a long-memory time series. While optimal block sizes are more complex under long-range dependence compared to short-range, we provide a consistent data-driven rule for block selection. Numerical studies illustrate that the guides for block selection perform well in other block-based problems with long-memory time series, such as distribution estimation and strategies for testing Hermite rank.

Funding Statement

Research was supported by NSF Grants DMS-1811998, DMS 2015390, and DMS-2131233.


The authors are grateful to two anonymous referees and an Associate Editor for constructive comments that improved the quality of this paper.


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Qihao Zhang. Soumendra N. Lahiri. Daniel J. Nordman. "On optimal block resampling for Gaussian-subordinated long-range dependent processes." Ann. Statist. 50 (6) 3619 - 3646, December 2022.


Received: 1 June 2022; Revised: 1 September 2022; Published: December 2022
First available in Project Euclid: 21 December 2022

Digital Object Identifier: 10.1214/22-AOS2242

Primary: 62G09
Secondary: 62G20 , 62M10

Keywords: block bootstrap , Block length , long-memory , subsampling , variance estimation

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 6 • December 2022
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