The Annals of Statistics

Scrambled net variance for integrals of smooth functions

Art B. Owen

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Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. One version, randomized $(t, m, s)$-nets, has the property that the integral estimates are unbiased and that the variance is $o(1/n)$, for any square integrable integrand.

Stronger assumptions on the integrand allow one to find rates of convergence. This paper shows that for smooth integrands over s dimensions, the variance is of order $n^{-3}(\log n)^{s-1}$, compared to $n^{-1}$ for ordinary Monte Carlo. Thus the integration errors are of order $n^{-3/2}(\log n)^{(s-1)/2} in probability. This compares favorably with the rate $n^{-1}(\log n)^{s-1}$ for unrandomized $(t, m, s)$-nets.

Article information

Ann. Statist. Volume 25, Number 4 (1997), 1541-1562.

First available in Project Euclid: 9 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods 68O20
Secondary: 62D05: Sampling theory, sample surveys 62K05: Optimal designs

Integration Latin hypercube multiresolution orthogonal array sampling quasi-Monte Carlo wavelets


Owen, Art B. Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 (1997), no. 4, 1541--1562. doi:10.1214/aos/1031594731.

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