On the dependence structure and quality of scrambled (t, m, s)-nets

Wiart Jaspar (), Lemieux Christiane () and Dong Gracia Y. ()
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Wiart Jaspar: Johannes Kepler University, Altenbergerstr. 69, 4040Linz, Austria
Lemieux Christiane: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, CanadaN2L 3G1
Dong Gracia Y.: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, CanadaN2L 3G1

Monte Carlo Methods and Applications, 2021, vol. 27, issue 1, 1-26

Abstract: In this paper we develop a framework to study the dependence structure of scrambled (t,m,s){(t,m,s)}-nets. It relies on values denoted by Cb⁢(𝒌;Pn){C_{b}({\boldsymbol{k}};P_{n})}, which are related to how many distinct pairs of points from Pn{P_{n}} lie in the same elementary 𝒌{{\boldsymbol{k}}}-interval in base b. These values quantify the equidistribution properties of Pn{P_{n}} in a more informative way than the parameter t. They also play a key role in determining if a scrambled set P~n{\widetilde{P}_{n}} is negative lower orthant dependent (NLOD). Indeed, this property holds if and only if Cb⁢(𝒌;Pn)≤1{C_{b}({\boldsymbol{k}};P_{n})\leq 1} for all 𝒌∈ℕs{{\boldsymbol{k}}\in\mathbb{N}^{s}}, which in turn implies that a scrambled digital (t,m,s){(t,m,s)}-net in base b is NLOD if and only if t=0{t=0}. Through numerical examples we demonstrate that these Cb⁢(𝒌;Pn){C_{b}({\boldsymbol{k}};P_{n})} values are a powerful tool to compare the quality of different (t,m,s){(t,m,s)}-nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.

Keywords: Negative dependence; scrambled nets; variance; quasi-Monte Carlo (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1515/mcma-2020-2079

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