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Hilbert space analysis of Latin Hypercube Sampling

Author: Peter Mathé
Journal: Proc. Amer. Math. Soc. 129 (2001), 1477-1492
MSC (2000): Primary 65C05; Secondary 62D05
Published electronically: October 24, 2000
MathSciNet review: 1814176
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Abstract: Latin Hypercube Sampling is a specific Monte Carlo estimator for numerical integration of functions on ${\mathbb R}^{d}$ with respect to some product probability distribution function. Previous analysis established that Latin Hypercube Sampling is superior to independent sampling, at least asymptotically; especially, if the function to be integrated allows a good additive fit. We propose an explicit approach to Latin Hypercube Sampling, based on orthogonal projections in an appropriate Hilbert space, related to the ANOVA decomposition, which allows a rigorous error analysis. Moreover, we indicate why convergence cannot be uniformly superior to independent sampling on the class of square integrable functions. We establish a general condition under which uniformity can be achieved, thereby indicating the rôle of certain Sobolev spaces.

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Peter Mathé
Affiliation: Weierstrass–Institute for Applied Analysis and Stochastics, Mohrenstraße 39, D– 10117 Berlin, Germany

Keywords: Latin Hypercube Sampling, stratified sampling, asymptotic variance
Received by editor(s): August 25, 1999
Published electronically: October 24, 2000
Communicated by: David Sharp
Article copyright: © Copyright 2000 American Mathematical Society