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Let $f:[0,1]^d\to\mathbb{R}$ be a completely monotone integrand as defined byAistleitner and Dick (2015) and let points$\boldsymbol{x}_0,\dots,\boldsymbol{x}_{n-1}\in[0,1]^d$ have a non-negativelocal discrepancy (NNLD) everywhere in $[0,1]^d$. We show how to use theseproperties to get a non-asymptotic and computable upper bound for the integralof $f$ over $[0,1]^d$. An analogous non-positive local discrepancy (NPLD)property provides a computable lower bound. It has been known since Gabai(1967) that the two dimensional Hammersley points in any base $b\ge2$ havenon-negative local discrepancy. Using the probabilistic notion of associatedrandom variables, we generalize Gabai's finding to digital nets in any base$b\ge2$ and any dimension $d\ge1$ when the generator matrices are permutationmatrices. We show that permutation matrices cannot attain the best values ofthe digital net quality parameter when $d\ge3$. As a consequence the computableabsolutely sure bounds we provide come with less accurate estimates than theusual digital net estimates do in high dimensions. We are also able toconstruct high dimensional rank one lattice rules that are NNLD. We show thatthose lattices do not have good discrepancy properties: any lattice rule withthe NNLD property in dimension $d\ge2$ either fails to be projection regular orhas all its points on the main diagonal.

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