Expected $L_2-$discrepancy bound for a class of new stratified sampling models

TL;DR

This paper introduces a new class of convex partitions for stratified sampling that achieve lower expected $L_2$-discrepancy than traditional methods, providing explicit bounds and identifying optimal partitions.

Contribution

It presents a novel class of convex equivolume partitions that improve the expected $L_2$-discrepancy bounds in stratified sampling, with explicit bounds and optimal configurations.

Findings

New convex partitions yield lower expected $L_2$-discrepancy than jittered sampling.

Explicit upper bounds for expected $L_2$-discrepancy are derived.

An optimal partition with the best expected $L_2$-discrepancy bound is identified.

Abstract

We introduce a class of convex equivolume partitions. Expected $L_2-$discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate random point sets with smaller expected $L_2-$discrepancy than classical jittered sampling for the same sampling number. Second, an explicit expected $L_2-$discrepancy upper bound under this kind of partitions is also given. Further, among these new partitions, there is optimal expected $L_2-$discrepancy upper bound.