Variants of the Koksma-Hlawka inequality for vertex-modified quasi-Monte Carlo integration rules

Vertex-modified rules have recently been introduced by the authors as a way of improving the performance of quasi-Monte Carlo methods for numerical integration. In this paper, we establish variants of the Koksma-Hlawka inequality for vertex-modified rules, and we show that there are choices for the vertex weights which, in general, yield smaller error bounds than the classical Koksma-Hlawka bound. Low-discrepancy point sets for which the local discrepancy has constant sign emerge as interesting objects of study.