Local antithetic sampling with scrambled nets
Art B. Owen
Source: Ann. Statist. Volume 36, Number 5 (2008), 2319-2343.
Abstract
We consider the problem of computing an approximation to the integral I=∫[0, 1]d f(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.
Primary Subjects: 65C05
Secondary Subjects: 68U20, 65D32
Keywords: Digital nets; monomial rules; randomized quasi-Monte Carlo; quasi-Monte Carlo
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1223908094
Digital Object Identifier: doi:10.1214/07-AOS548
Mathematical Reviews number (MathSciNet):
MR2458189
Zentralblatt MATH identifier:
1157.65006
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