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

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.