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 $\mathbb{R}^{s-1}$, the inverse Rosenblatt transformation can be used to convert samples from the $(s-1)$-dimensional cube to $\mathbb{R}^{s-1}$. 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 $\mathbb{R}^{s-1}$. 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.]
Highlights
1.1 Scope of researchNumerical integration is a common computational problem occurring in many areas of science, such as statistics, financial mathematics and computational physics, where one has to compute some integral, for instance an expectation value, which cannot be done analytically
We prove a convergence rate of order N ↵ for 1/s ↵ < 1, where ↵ depends on the target density, more explicitly, where the value of ↵ here depends on how well the graph of the target density can be covered by certain rectangles
In Section 3.2.4, we proved a convergence rate of order N ↵ for 1/s ↵ < 1 for samples generated by an acceptance-rejection sampler using (t, m, s)-nets as a driver sequence, where ↵ depends on the target density
Summary
2.1 Local discrepancy of points in the rectangle [0, t ) ⇥ [0, t ). 2.2 The discrepancy measures the di↵erence between the proportion of points in each rectangle J which is anchored at the origin and the Lebesgue measure of J. The star-discrepancy is defined by the supremum of the local discrepancy function over all anchored rectangles J. J to be all rectangles [ , ), we obtain the so-called extreme discrepancy.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
