"On Reduction of Finite Sample Variance by Extended Latin Hypercube Sampling"

McKay, Conover and Beckman (1979) introduced Latin hypercube sampling (LHS) for reducing variance of Monte Carlo simulations. More recently Owen (1992a) and Tang (1993) generalized LHS using orthogonal arrays. In the Owen's class of generalized LHS, we define extended Latin hypercube sampling of strength m (henceforth denoted as ELHS(m)), such that ELHS(1) reduces to LHS. We first derive explicit formula for the finite sample variance of ELHS(m) by detailed investigation of combinatories involved in ELHS(m). Based on this formula, we give a sufficient condition for variance reduction by ELHS(m), generalizing similar result of McKay, Conover and Beckman (1979) for m = 1. Actually our sufficient condition for m=1 contains the sufficient condition by McKay, Conover and Beckman (1979) and thus strengthens their result.