Lattice Sampling Revisited: Monte Carlo Variance of Means Over Randomized Orthogonal Arrays

Abstract

Randomized orthogonal arrays provide good sets of input points for exploration of computer programs and for Monte Carlo integration. In 1954, Patterson gave a formula for the randomization variance of the sample mean of a function evaluated at the points of an orthogonal array. That formula is incorrect for most of the arrays that are practical for computer experiments. In this paper we correct Patterson's formula. We also remark on a defect, related to coincidences, in some orthogonal arrays. These are arrays of the form $OA(2q^2, 2q + 1, q, 2)$, where $q$ is a prime power, obtained by constructions due to Bose and Bush and to Addelman and Kempthorne. We conjecture that subarrays of the form $OA(2q^2, 2q, q, 2)$ may be constructed to avoid this defect.