Electronic Journal of Statistics

Discrepancy bounds for deterministic acceptance-rejection samplers

Houying Zhu and Josef Dick

Full-text: Open access

Abstract

We consider an acceptance-rejection sampler based on a deterministic driver sequence. The deterministic sequence is chosen such that the discrepancy between the empirical target distribution and the target distribution is small. We use quasi-Monte Carlo (QMC) point sets for this purpose. The empirical evidence shows convergence rates beyond the crude Monte Carlo rate of $N^{-1/2}$. We prove that the discrepancy of samples generated by the QMC acceptance-rejection sampler is bounded from above by $N^{-1/s}$. A lower bound shows that for any given driver sequence, there always exists a target density such that the star discrepancy is at most $N^{-2/(s+1)}$. For a general density, whose domain is the real state space $\mathbb{R}^{s-1}$, the inverse Rosenblatt transformation can be used to convert samples from the $(s-1)$-dimensional cube to $\mathbb{R}^{s-1}$. We show that this transformation is measure preserving. This way, under certain conditions, we obtain the same convergence rate for a general target density defined in $\mathbb{R}^{s-1}$. Moreover, we also consider a deterministic reduced acceptance-rejection algorithm recently introduced by Barekat and Caflisch [F. Barekat and R. Caflisch, Simulation with Fluctuation and Singular Rates. ArXiv:1310.4555 [math.NA], 2013.]

Article information

Source
Electron. J. Statist. Volume 8, Number 1 (2014), 678-707.

Dates
First available in Project Euclid: 21 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1400703410

Digital Object Identifier
doi:10.1214/14-EJS898

Mathematical Reviews number (MathSciNet)
MR3211028

Zentralblatt MATH identifier
1348.60113

Subjects
Primary: 62F15: Bayesian inference
Secondary: 11K45: Pseudo-random numbers; Monte Carlo methods

Keywords
Acceptance-rejection sampler Star discrepancy $(t,m,s)$-nets

Citation

Zhu, Houying; Dick, Josef. Discrepancy bounds for deterministic acceptance-rejection samplers. Electron. J. Statist. 8 (2014), no. 1, 678--707. doi:10.1214/14-EJS898. https://projecteuclid.org/euclid.ejs/1400703410.


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