Discrepancy estimates for acceptance-rejection samplers using stratified inputs

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 −s)-dimensions generated by this algorithm. First we show an upper bound on the star-discrepancy of order N−d/2−1/(2s). Further we prove an upper bound on the qth moment of the Lq-discrepancy (E[Nq Lqq,N])1/q for 2 ≤ q ≤ ∞, 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.