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.

翻译：我们研究了一类新设计的非等体积分割下的期望星偏差。主要贡献有两个方面。首先，我们为星偏差建立了一个强分割原理，表明新设计的非等体积分割产生的分层抽样点集比经典抖动抽样具有更低的期望星偏差。具体而言，我们证明了 $\mathbb{E}(D^{*}_{N}(Z)) < \mathbb{E}(D^{*}_{N}(Y))$，其中 $Y$ 和 $Z$ 分别代表抖动抽样和我们的非等体积分割抽样。其次，我们在非等体积分割模型下推导了期望星偏差的显式上界，这些上界改进了现有抖动抽样的界。我们的结果为在高维数值积分中使用非等体积分割提供了理论基础。