Scrambled net variance for integrals of smooth functions
type: Item
instance of: scholarly article
title: Scrambled net variance for integrals of smooth functions
zbMATH Open document ID: 0886.65018
author: Art B. Owen
published in: The Annals of Statistics
publication date: 1998-02-18T00:00:00Z
review text: Hybrids of quasi-Monte Carlo and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the later. This paper studies the variance of one such hybrid, randomized \((t,m,s)\)-nets, one of the best low-discrepancy sequence, by applying a multidimensional multiresolution (wavelet) analysis to the integrand. For any square integrable integrand over \(s\) dimensions, the integral estimates are unbiased and the variance is \(o(1/n)\). For smooth integrand, the variance is even of order \(n^{-3}n^{-3}(\log n)^{s-1}\), compared to \(n^{-1}\) for classical Monte Carlo method. Thus the integration errors are of order \(n^{-3/2}(\log n)^{(s-1)/2}\) in probability which compares favorably with the rate \(n^{-1}(\log n)^{s-1}\) for the best low-discrepancy sequence. Of course, the rate for randomized \((t,m,s)\)-nets is an average case result for a fixed function, while the rate for the latter describes the worst case over functions, for a fixed set of integration points.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65C05
zbMATH DE Number: 1082920
zbMATH Keywords: integration
zbMATH Keywords: Latin hypercube
zbMATH Keywords: error estimation methods
zbMATH Keywords: orthogonal array sampling
zbMATH Keywords: randomized \((t,m,s)\)-net
zbMATH Keywords: multiresolution (wavelet) analysis
zbMATH Keywords: quasi-Monte Carlo methods
describes a project that uses: TESTPACK
reviewed by: Y.-J. Xiao
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1031594731
OpenAlex ID: W2024626809
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DOI: 10.1214/AOS/1031594731
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description: scientific article; zbMATH DE number 1082920