Hilbert space analysis of Latin Hypercube Sampling

Latin Hypercube Sampling is a specific Monte Carlo estimator for numerical integration of functions on R-d with respect to some product probability di stribution function. Previous analysis established that Latin Hypercube Sam pling is superior to independent sampling, at least asymptotically; especia lly, if the function to be integrated allows a good additive fit. We propos e an explicit approach to Latin Hypercube Sampling, based on orthogonal pro jections in an appropriate Hilbert space, related to the ANOVA decompositio n, which allows a rigorous error analysis. Moreover, we indicate why conver gence cannot be uniformly superior to independent sampling on the class of square integrable functions. We establish a general condition under which u niformity can be achieved, thereby indicating the role of certain Sobolev s paces.