Expected star discrepancy based on stratified sampling

TL;DR

This paper improves theoretical bounds on star discrepancy for stratified sampling methods and proves that stratified sampling consistently outperforms random sampling in expected discrepancy, confirmed by numerical simulations.

Contribution

It derives a sharper expected upper bound for jittered sampling and proves the strong partition principle, showing stratified sampling's superiority over random sampling.

Findings

Sharper expected upper bound for jittered sampling

Stratified sampling yields smaller expected discrepancy than random sampling

Numerical simulations confirm theoretical results

Abstract

We present two main contributions to the expected star discrepancy theory. First, we derive a sharper expected upper bound for jittered sampling, improving the leading constants and logarithmic terms compared to the state-of-the-art [Doerr, 2022]. Second, we prove the strong partition principle for star discrepancy, showing that any equal-measure stratified sampling yields a strictly smaller expected discrepancy than simple random sampling, thereby resolving an open question in [Kiderlen and Pausinger, 2022]. Numerical simulations confirm our theoretical advances and illustrate the superiority of stratified sampling in low to moderate dimensions.