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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Wiart Jaspar & Lemieux Christiane & Dong Gracia Y., 2021. "On the dependence structure and quality of scrambled (t, m, s)-nets," Monte Carlo Methods and Applications, De Gruyter, vol. 27(1), pages 1-26, March.
    • Handle: RePEc:bpj:mcmeap:v:27:y:2021:i:1:p:1-26:n:5
    DOI: 10.1515/mcma-2020-2079
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    References listed on IDEAS

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    1. Christiane Lemieux, 2018. "Negative Dependence, Scrambled Nets, and Variance Bounds," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 228-251, February.
    2. Faure, Henri & Lemieux, Christiane, 2019. "Implementation of irreducible Sobol’ sequences in prime power bases," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 161(C), pages 13-22.
    Full references (including those not matched with items on IDEAS)

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