Discrepancy bounds for a deterministic acceptance-rejection sampler

Abstract: 

Markov chain Monte Carlo(MCMC) is a staple tool in statistics used for sampling from a distribution where direct simulation is not possible (or difficult). In this talk we consider  an acceptance-rejection sampler based on a deterministic driver sequence, which is a simple case of a Markov chain quasi-Monte Carlo (MCQMC) method. The deterministic sequence is chosen such that the discrepancy between the empirical proposal distribution and the proposal distribution is small. We use quasi-Monte Carlo (QMC) point sets for this purpose. In a general setting, we prove an upper bound and a lower bound of the discrepancy of samples generated by the deterministic acceptance-rejection sampler. Improved upper bounds are obtained under certain assumptions.  The numerical experiments verify the theoretical conclusions and show convergence rates beyond plain Monte Carlo.