The Annals of Statistics

Local antithetic sampling with scrambled nets

Art B. Owen

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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.

Article information

Ann. Statist., Volume 36, Number 5 (2008), 2319-2343.

First available in Project Euclid: 13 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods
Secondary: 68U20: Simulation [See also 65Cxx] 65D32: Quadrature and cubature formulas

Digital nets monomial rules randomized quasi-Monte Carlo quasi-Monte Carlo


Owen, Art B. Local antithetic sampling with scrambled nets. Ann. Statist. 36 (2008), no. 5, 2319--2343. doi:10.1214/07-AOS548.

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