On the distribution of scrambled $(0,m,s)$-nets over unanchored boxes

TL;DR

This paper introduces a new measure called the pairwise sampling dependence index to evaluate the quality of randomized low-discrepancy point sets, showing scrambled $(0,m,s)$-nets have desirable negative dependence properties for integration over unanchored boxes.

Contribution

The paper proposes a novel quality measure based on negative dependence and demonstrates that scrambled $(0,m,s)$-nets possess this property, improving integration variance.

Findings

Scrambled $(0,m,s)$-nets have a negative pairwise sampling dependence index.

Digital shift randomization may result in a positive index.

The new measure assesses the variance reduction in integration tasks.

Abstract

We introduce a new quality measure to assess randomized low-discrepancy point sets of finite size $n$. This new quality measure, which we call "pairwise sampling dependence index", is based on the concept of negative dependence. A negative value for this index implies that the corresponding point set integrates the indicator function of any unanchored box with smaller variance than the Monte Carlo method. We show that scrambled $(0,m,s)-$nets have a negative pairwise sampling dependence index. We also illustrate through an example that randomizing via a digital shift instead of scrambling may yield a positive pairwise sampling dependence index.