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• W3125876763 abstract "Abstract In this paper we develop a framework to study the dependence structure of scrambled <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:math> {(t,m,s)} -nets. It relies on values denoted by <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>b</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝒌</m:mi> <m:mo>;</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {C_{b}({boldsymbol{k}};P_{n})} , which are related to how many distinct pairs of points from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>P</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {P_{n}} lie in the same elementary <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>𝒌</m:mi> </m:math> {{boldsymbol{k}}} -interval in base b . These values quantify the equidistribution properties of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>P</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {P_{n}} in a more informative way than the parameter t . They also play a key role in determining if a scrambled set <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mover accent=true> <m:mi>P</m:mi> <m:mo>~</m:mo> </m:mover> <m:mi>n</m:mi> </m:msub> </m:math> {widetilde{P}_{n}} is negative lower orthant dependent (NLOD). Indeed, this property holds if and only if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>b</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝒌</m:mi> <m:mo>;</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {C_{b}({boldsymbol{k}};P_{n})leq 1} for all <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>𝒌</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℕ</m:mi> <m:mi>s</m:mi> </m:msup> </m:mrow> </m:math> {{boldsymbol{k}}inmathbb{N}^{s}} , which in turn implies that a scrambled digital <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:math> {(t,m,s)} -net in base b is NLOD if and only if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {t=0} . Through numerical examples we demonstrate that these <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>b</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝒌</m:mi> <m:mo>;</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {C_{b}({boldsymbol{k}};P_{n})} values are a powerful tool to compare the quality of different <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:math> {(t,m,s)} -nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets." @default.
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• W3125876763 date "2021-01-10" @default.
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• W3125876763 title "On the dependence structure and quality of scrambled (<i>t</i>, <i>m</i>, <i>s</i>)-nets" @default.
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