Computable error bounds for quasi-Monte Carlo using points with non-negative local discrepancy

Abstract

Abstract Let $f:[0,1]^{d}\to{\mathbb{R}}$ be a completely monotone integrand as defined by Aistleitner and Dick (2015, Acta Arithmetica, 167, 143–171) and let points $\boldsymbol{x}_{0},\dots ,\boldsymbol{x}_{n-1}\in [0,1]^{d}$ have a non-negative local discrepancy (NNLD) everywhere in $[0,1]^{d}$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $f$ over $[0,1]^{d}$. An analogous non-positive local discrepancy property provides a computable lower bound. It has been known since Gabai (1967, Illinois J. Math., 11, 1–12) that the two-dimensional Hammersley points in any base $b\geqslant 2$ have NNLD. Using the probabilistic notion of associated random variables, we generalize Gabai’s finding to digital nets in any base $b\geqslant 2$ and any dimension $d\geqslant 1$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $d\geqslant 3$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high-dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $d\geqslant 2$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate.