Unbiased Monte Carlo integration methods with exactness for low order polynomials

We consider methods for estimation of the integral of a given function that combine the unbiasedness of Monte Carlo integration (which permits a simple statistical assessment of the error) with the higher precision often attained by deterministic methods. We propose symmetric random designs with $2k...

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Bibliographic Details
Published in:SIAM journal on scientific and statistical computing 1985, Vol.6 (1), p.169-181
Main Authors: SIEGEL, A. F, O'BRIEN, F
Format: Article
Language:English
Subjects:
Accuracy
Applied mathematics
Estimates
Exact sciences and technology
Investigations
Mathematical functions
Mathematics
Methods
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Polynomials
Random variables
Sciences and techniques of general use
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Summary:We consider methods for estimation of the integral of a given function that combine the unbiasedness of Monte Carlo integration (which permits a simple statistical assessment of the error) with the higher precision often attained by deterministic methods. We propose symmetric random designs with $2k + 1$ points that achieve exactness for polynomials of degree up to $2k + 1$. The distribution for the three-point method is unique, although for higher order methods there are multiple choices for the sampling distribution. For two-dimensional multiple integration over a rectangle, we propose an unbiased five-point method that achieves exactness for polynomials of degree up to three in both variables. Some bounds on error variances are given.
ISSN:0196-5204
1064-8275
2168-3417
1095-7197
DOI:10.1137/0906014