Star discrepancy for new stratified random sampling I: optimal expected star discrepancy

TL;DR

This paper introduces a new class of convex partitions for stratified sampling, identifying optimal partitions that minimize expected star discrepancy and outperform jittered sampling, thus advancing discrepancy theory and sampling methods.

Contribution

It identifies partitions that minimize expected star discrepancy, generalizes discrepancy bounds, and solves open questions in stratified sampling theory.

Findings

One partition minimizes expected star discrepancy.

Infinite partitions outperform classical jittered sampling.

Optimal discrepancy bounds are established for these partitions.

Abstract

We introduce a class of convex equivolume partitions. Expected star discrepancy results are compared for stratified samples under these partitions, including simple random samples. There are four main parts of our results. First, among these newly designed partitions, there is one that minimizes the expected star discrepancy, thus we partly answer an open question in [F. Pausinger, S. Steinerberger, J. Complex. 2016]. Second, there are an infinite number of such class of partitions, which generate point sets with smaller expected discrepancy than classical jittered sampling for large sampling number, leading to an open question in [M. Kiderlen, F. Pausinger, Monatsh. Math. 2021] being solved. Third, we prove a strong partition principle and generalize the expected star discrepancy under these partition models from $L_2-$discrepancy to star discrepancy, hence an open question in [M. Kiderlen, F. Pausinger, J. Complex. 2021] is answered. In the end, optimal expected star discrepancy upper bound under this class of partitions is given, which is better than using jittered sampling.