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Negative association, ordering and convergence of resampling methods

Author

Listed:
  • Mathieu GERBER

    (School of Mathematics, University of Bristol)

  • Nicolas CHOPIN

    (CREST-ENSAE)

  • Nick WHITELEY

    (School of Mathematics, University of Bristol)

Abstract

We study convergence and convergence rates for resampling. Our ?rst 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 (1996) strati?ed resampling method. Carpenter et al’s (1999) 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 Srinivasan (2001), 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 con?rm a conjecture made by Kitagawa (1996) 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 Rd, 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 di?er from multinomial resampling.

Suggested Citation

  • Mathieu GERBER & Nicolas CHOPIN & Nick WHITELEY, 2017. "Negative association, ordering and convergence of resampling methods," Working Papers 2017-36, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2017-36
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    References listed on IDEAS

    as
    1. Zhijian He & Art B. Owen, 2016. "Extensible grids: uniform sampling on a space filling curve," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 917-931, September.
    2. Mathieu Gerber & Nicolas Chopin, 2015. "Sequential quasi Monte Carlo," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(3), pages 509-579, June.
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