Hilbert space analysis of Latin Hypercube Sampling

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.