Abstract
In this paper we propose an acceptance-rejection sampler using stratified inputs as driver sequence. We estimate the discrepancy of the N-point set in \((s-1)\)-dimensions generated by this algorithm. First we show an upper bound on the star-discrepancy of order \(N^{-1/2-1/(2s)}\). Further we prove an upper bound on the qth moment of the \(L_q\)-discrepancy \((\mathbb {E}[N^{q}L^{q}_{q,N}])^{1/q}\) for \(2\le q\le \infty \), which is of order \(N^{(1-1/s)(1-1/q)}\). The proposed approach is numerically tested and compared with the standard acceptance-rejection algorithm using pseudo-random inputs. We also present an improved convergence rate for a deterministic acceptance-rejection algorithm using \((t,m,s)-\)nets as driver sequence.
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Acknowledgments
The work was supported by Australian Research Council Discovery Project DP150101770. We thank Daniel Rudolf and the anonymous referee for many very helpful comments.
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Zhu, H., Dick, J. (2016). Discrepancy Estimates For Acceptance-Rejection Samplers Using Stratified Inputs. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_33
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