Open Access
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


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.


Download Citation

Mathieu Gerber. Nicolas Chopin. Nick Whiteley. "Negative association, ordering and convergence of resampling methods." Ann. Statist. 47 (4) 2236 - 2260, August 2019.


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

Primary: 62G09 , 62M20
Secondary: 60G35

Keywords: negative association , particle filtering , Resampling methods

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 4 • August 2019
Back to Top