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August 1997 Scrambled net variance for integrals of smooth functions
Art B. Owen
Ann. Statist. 25(4): 1541-1562 (August 1997). DOI: 10.1214/aos/1031594731


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.


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Art B. Owen. "Scrambled net variance for integrals of smooth functions." Ann. Statist. 25 (4) 1541 - 1562, August 1997.


Published: August 1997
First available in Project Euclid: 9 September 2002

zbMATH: 0886.65018
MathSciNet: MR1463564
Digital Object Identifier: 10.1214/aos/1031594731

Primary: 65C05, 68O20
Secondary: 62D05, 62K05

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 4 • August 1997
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