The Annals of Statistics

Scrambled net variance for integrals of smooth functions

Art B. Owen

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 9 September 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1031594731

Mathematical Reviews number (MathSciNet)
MR1463564

Digital Object Identifier
doi:10.1214/aos/1031594731

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

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

Citation

Owen, Art B. Scrambled net variance for integrals of smooth functions. The Annals of Statistics 25 (1997), no. 4, 1541--1562. doi:10.1214/aos/1031594731. http://projecteuclid.org/euclid.aos/1031594731.


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  • STANFORD, CALIFORNIA 94305 E-MAIL: owen@play fair.stanford.edu