## Abstract

In this paper we consider an acceptance-rejection (AR) sampler based on deterministic driver sequences. We prove that the discrepancy of an N element sample set generated in this way is bounded by O(N^{-2/3}log N) , provided that the target density is twice continuously differentiable with non-vanishing curvature and the AR sampler uses the driver sequence KM={(jα,jβ)mod1∣j=1,…,M}, where α, β are real algebraic numbers such that 1, α, β is a basis of a number field over Q of degree 3. For the driver sequence F_{k}= { (j/F_{k}, { jF_{k}_{-}_{1}/F_{k}}) ∣ j=1 , … , F_{k}} , where F_{k} is the k-th Fibonacci number and {x} = x- ⌊x⌋ is the fractional part of a non-negative real number x, we can remove the log factor to improve the convergence rate to O(N^{-2/3}) , where again N is the number of samples we accepted. We also introduce a criterion for measuring the goodness of driver sequences. The proposed approach is numerically tested by calculating the star-discrepancy of samples generated for some target densities using K_{M} and F_{k} as driver sequences. These results confirm that achieving a convergence rate beyond N^{-1/2} is possible in practice using K_{M} and F_{k} as driver sequences in the acceptance-rejection sampler.

Original language | English |
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Pages (from-to) | 901-911 |

Number of pages | 11 |

Journal | Statistics and Computing |

Volume | 27 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2017 |

Externally published | Yes |

## Keywords

- Acceptance-rejection sampler
- Discrepancy
- Fibonacci lattice points
- Integration error