Papers › On the dependence structure and quality of scrambled (t,m,s)-nets
On the dependence structure and quality of scrambled (t,m,s)-nets
Jaspar Wiart, Christiane Lemieux, Gracia Y. Dong
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In this paper we develop a framework to study the dependence structure of scrambled (t,m,s)-nets. It relies on values denoted by C_b(𝐤;Pₙ), which are related to how many distinct pairs of points from Pₙ lie in the same elementary 𝐤-interval in base b. These values quantify the equidistribution properties of Pₙ in a more informative way than the parameter t. They also play a key role in determining if a scrambled set P̃ₙ is negative lower orthant dependent (NLOD). Indeed this property holds if and only if C_b(𝐤;Pₙ) ≤1 for all 𝐤 ∈ℕˢ, which in turn implies that a scrambled digital $(t,m,s)-$net in base b is NLOD if and only if t=0. Through numerical examples we demonstrate that these C_b(𝐤;Pₙ) values are a powerful tool to compare the quality of different (t,m,s)-nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.
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