On a partition with a lower expected $\mathcal{L}_2$-discrepancy than classical jittered sampling

TL;DR

This paper demonstrates that classical jittered sampling is not optimal for minimizing expected -discrepancy in stratified sampling, providing an explicit counterexample with convex equal-volume partitions.

Contribution

It proves that alternative stratified partitions can achieve lower expected -discrepancy than classical jittered sampling, challenging existing assumptions.

Findings

Classical jittered sampling does not minimize expected -discrepancy.

Explicit counterexamples with convex equal-volume partitions are constructed.

Alternative stratified partitions can outperform jittered sampling.

Abstract

We prove that classical jittered sampling of the $d$-dimensional unit cube does not yield the smallest expected $\mathcal{L}_2$-discrepancy among all stratified samples with $N=m^d$ points. Our counterexample can be given explicitly and consists of convex partitioning sets of equal volume.