Note on Pairwise Negative Dependence of Randomized Rank-1 Lattices
For researchers in quasi-Monte Carlo methods, this provides a new class of point sets with guaranteed variance reduction for monotone functions in any dimension, though the result is incremental as it extends existing theory.
This note shows that randomly shifted and jittered rank-1 lattices in arbitrary dimension d satisfy a pairwise negative dependence property, which implies that randomized quasi-Monte Carlo estimators using these point sets have variance no larger than Monte Carlo estimators for monotone functions. The result extends a previous finding for (0,m,2)-nets to higher dimensions, but the specific randomization is critical as small changes can break the property.
In her recent paper [Negative dependence, scrambled nets, and variance bounds. Math. Oper. Res. 43 (2018), 228-251] Christiane Lemieux studied a framework to analyze the dependence structure of sampling schemes. The main goal of the framework is to determine conditions under which the negative dependence structure of a sampling scheme yields estimators with reduced variance compared to Monte Carlo estimators. For instance, she was able to show that in dimension d = 2 scrambled (0,m, d)-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. Her result relies on a pairwise negative dependence property that is, in particular, satisfied by (0,m, 2)-nets. In this note we establish that the same result holds true in arbitrary dimension d for a type of randomized lattice point sets that we call randomly shifted and jittered rank-1 lattices. We show that the details of the randomization are crucial and that already small modifications may destroy the pairwise negative dependence property.