The Partition Principle Revisited: Non-Equal Volume Designs Achieve Minimal Expected Star Discrepancy

Abstract: We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed non-equal volume partitions yield stratified sampling point sets with lower expected star discrepancy than classical jittered sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z)) < \mathbb{E}(D^{*}_{N}(Y))$, where $Y$ and $Z$ represent jittered sampling and our non-equal volume partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our non-equal volume partition models, which improve upon existing bounds for jittered sampling. Our results provide a theoretical foundation for using non-equal volume partitions in high-dimensional numerical integration.