## Hilbert space analysis of Latin Hypercube Sampling

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- Proc. Amer. Math. Soc.
**129**(2001), 1477-1492 Request permission

## 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.## References

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## Additional Information

**Peter Mathé**- Affiliation: Weierstrass–Institute for Applied Analysis and Stochastics, Mohrenstraße 39, D– 10117 Berlin, Germany
- Email: mathe@wias-berlin.de
- Received by editor(s): August 25, 1999
- Published electronically: October 24, 2000
- Communicated by: David Sharp
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**129**(2001), 1477-1492 - MSC (2000): Primary 65C05; Secondary 62D05
- DOI: https://doi.org/10.1090/S0002-9939-00-05850-0
- MathSciNet review: 1814176