Journal Article

NEGATIVE ASSOCIATION, ORDERING AND CONVERGENCE OF RESAMPLING METHODS

Gerber, Mathieu, Whiteley, Nick, Chopin, Nicolas

2019

Overview

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 ℝ yields a faster rate of convergence; we establish that when particles are ordered using the Hilbert curve in ℝ d , the variance of the resampling error is 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.

• Publisher: Institute of Mathematical Statistics
• Subject: Algorithms / Asymptotic methods / Asymptotic properties / Convergence / Hilbert curve / Mathematics / Resampling / Rounding / Sampling / Theorems