### 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 ℝ^{s-1}, the inverse Rosenblatt transformation can be used to convert samples from the (s-1)-dimensional cube to 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 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].

Original language | English |
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Pages (from-to) | 678-707 |

Number of pages | 30 |

Journal | Electronic Journal of Statistics |

Volume | 8 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

### Keywords

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

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## Cite this

*Electronic Journal of Statistics*,

*8*(1), 678-707. https://doi.org/10.1214/14-EJS898