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August 2019 Negative association, ordering and convergence of resampling methods
Mathieu Gerber, Nicolas Chopin, Nick Whiteley
Ann. Statist. 47(4): 2236-2260 (August 2019). DOI: 10.1214/18-AOS1746

Abstract

We study convergence and convergence rates for resampling schemes. Our first main result is a general consistency theorem based on the notion of negative association, which is applied to establish the almost sure weak convergence of measures output from Kitagawa’s [J. Comput. Graph. Statist. 5 (1996) 1–25] stratified resampling method. Carpenter, Ckiffird and Fearnhead’s [IEE Proc. Radar Sonar Navig. 146 (1999) 2–7] systematic resampling method is similar in structure but can fail to converge depending on the order of the input samples. We introduce a new resampling algorithm based on a stochastic rounding technique of [In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001) (2001) 588–597 IEEE Computer Soc.], which shares some attractive properties of systematic resampling, but which exhibits negative association and, therefore, converges irrespective of the order of the input samples. We confirm a conjecture made by [J. Comput. Graph. Statist. 5 (1996) 1–25] that ordering input samples by their states in $\mathbb{R}$ yields a faster rate of convergence; we establish that when particles are ordered using the Hilbert curve in $\mathbb{R}^{d}$, the variance of the resampling error is ${\scriptstyle\mathcal{O}}(N^{-(1+1/d)})$ under mild conditions, where $N$ is the number of particles. We use these results to establish asymptotic properties of particle algorithms based on resampling schemes that differ from multinomial resampling.

Citation

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Mathieu Gerber. Nicolas Chopin. Nick Whiteley. "Negative association, ordering and convergence of resampling methods." Ann. Statist. 47 (4) 2236 - 2260, August 2019. https://doi.org/10.1214/18-AOS1746

Information

Received: 1 August 2017; Revised: 1 June 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082285
MathSciNet: MR3953450
Digital Object Identifier: 10.1214/18-AOS1746

Subjects:
Primary: 62G09, 62M20
Secondary: 60G35

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • August 2019
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