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 de ne extended Latin hypercube sampling of strengthm (henceforth denoted as ELHS(m)), such that ELHS(1) reduces to LHS. We rst derive explicit formula for the nite sample variance of ELHS(m) by detailed investigation of combinatorics involved in ELHS(m). Based on this formula, we give a su cient condition for variance reduction by ELHS(m), generalizing similar result of McKay, Conover and Beckman (1979) for m = 1. Actually our su cient condition for m = 1 contains the su cient condition by McKay, Conover and Beckman (1979) and thus strengthens their result.