Note on pairwise negative dependence of randomly shifted and jittered rank-1 lattices

Abstract

In a recent paper Christiane Lemieux analyzed pairwise negative dependence property of sampling schemes. Using it she showed that scrambled (0, m, 2)-nets lead to randomized quasi-Monte Carlo estimators with variance no larger than the variance of Monte Carlo estimators for functions monotone in each variable. We establish that the same result holds in arbitrary dimension for a specific randomized point set based on rank-1 lattices and show that the details of the randomization are crucial for the property to hold. (C) 2020 Elsevier B.V. All rights reserved.