On a Class of Partitions with Lower Expected Star Discrepancy and Its Upper Bound than Jittered Sampling

TL;DR

This paper introduces a new class of convex partitions that produce stratified sampling sets with lower expected star discrepancy than jittered and random sampling, providing explicit bounds and resolving an open question.

Contribution

The paper develops a novel partition model that improves star discrepancy bounds and establishes a strong partition principle, advancing stratified sampling theory.

Findings

New partition models outperform jittered sampling in expected star discrepancy.

Explicit upper bounds for star discrepancy are derived and are tighter than existing bounds.

The results answer an open question about the strong partition principle for star discrepancy.

Abstract

We investigate the expected star discrepancy under a newly designed class of convex equivolume partition models. The main contributions are two-fold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed partitions yield stratified sampling point sets with lower expected star discrepancy than both classical jittered sampling and simple random sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z))\leq\mathbb{E}(D^{*}_{N}(Y))<\mathbb{E}(D^{*}_{N}(X))$, where $X$, $Y$, and $Z$ represent simple random sampling, jittered sampling, and our new partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our partition models, which improve upon existing bounds for jittered sampling. Our results resolve Open Question 2 posed in Kiderlen and Pausinger (2021) regarding the strong partition principle for star discrepancy.