The Annals of Statistics

Convolved subsampling estimation with applications to block bootstrap

Johannes Tewes, Dimitris N. Politis, and Daniel J. Nordman

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The block bootstrap approximates sampling distributions from dependent data by resampling data blocks. A fundamental problem is establishing its consistency for the distribution of a sample mean, as a prototypical statistic. We use a structural relationship with subsampling to characterize the bootstrap in a new and general manner. While subsampling and block bootstrap differ, the block bootstrap distribution of a sample mean equals that of a $k$-fold self-convolution of a subsampling distribution. Motivated by this, we provide simple necessary and sufficient conditions for a convolved subsampling estimator to produce a normal limit that matches the target of bootstrap estimation. These conditions may be linked to consistency properties of an original subsampling distribution, which are often obtainable under minimal assumptions. Through several examples, the results are shown to validate the block bootstrap for means under significantly weakened assumptions in many existing (and some new) dependence settings, which also addresses a standing conjecture of Politis, Romano and Wolf [Subsampling (1999) Springer]. Beyond sample means, convolved subsampling may not match the block bootstrap, but instead provides an alternative resampling estimator that may be of interest. Under minimal dependence conditions, results also broadly establish convolved subsampling for general statistics having normal limits.

Article information

Ann. Statist., Volume 47, Number 1 (2019), 468-496.

Received: April 2017
Revised: February 2018
First available in Project Euclid: 30 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G09: Resampling methods
Secondary: 62G20: Asymptotic properties 62J05: Linear regression 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Convolution mixing moving blocks nonstationary


Tewes, Johannes; Politis, Dimitris N.; Nordman, Daniel J. Convolved subsampling estimation with applications to block bootstrap. Ann. Statist. 47 (2019), no. 1, 468--496. doi:10.1214/18-AOS1695.

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Supplemental materials

  • Supplement to “Convolved subsampling estimation with applications to block bootstrap”. This supplement provides proofs for the distributional results about convolved subsampling and further numerical/theoretical support for the simulation study.