Open Access
October 2008 Local antithetic sampling with scrambled nets
Art B. Owen
Ann. Statist. 36(5): 2319-2343 (October 2008). DOI: 10.1214/07-AOS548


We consider the problem of computing an approximation to the integral I=[0, 1]df(x) dx. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of O(n−1/2) from n independent random function evaluations. By contrast, quasi-Monte Carlo (QMC) sampling using carefully equispaced evaluation points can attain the rate O(n−1+ɛ) for any ɛ>0 and randomized QMC (RQMC) can attain the RMSE O(n−3/2+ɛ), both under mild conditions on f.

Classical variance reduction methods for MC can be adapted to QMC. Published results combining QMC with importance sampling and with control variates have found worthwhile improvements, but no change in the error rate. This paper extends the classical variance reduction method of antithetic sampling and combines it with RQMC. One such method is shown to bring a modest improvement in the RMSE rate, attaining O(n−3/2−1/d+ɛ) for any ɛ>0, for smooth enough f.


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Art B. Owen. "Local antithetic sampling with scrambled nets." Ann. Statist. 36 (5) 2319 - 2343, October 2008.


Published: October 2008
First available in Project Euclid: 13 October 2008

zbMATH: 1157.65006
MathSciNet: MR2458189
Digital Object Identifier: 10.1214/07-AOS548

Primary: 65C05
Secondary: 65D32 , 68U20

Keywords: Digital nets , monomial rules , quasi-Monte Carlo , randomized quasi-Monte Carlo

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 5 • October 2008
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