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2014 Discrepancy bounds for deterministic acceptance-rejection samplers
Houying Zhu, Josef Dick
Electron. J. Statist. 8(1): 678-707 (2014). DOI: 10.1214/14-EJS898


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.]


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Houying Zhu. Josef Dick. "Discrepancy bounds for deterministic acceptance-rejection samplers." Electron. J. Statist. 8 (1) 678 - 707, 2014.


Published: 2014
First available in Project Euclid: 21 May 2014

zbMATH: 1348.60113
MathSciNet: MR3211028
Digital Object Identifier: 10.1214/14-EJS898

Primary: 62F15
Secondary: 11K45

Rights: Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society


Vol.8 • No. 1 • 2014
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