On the dependence structure and quality of scrambled (<i>t</i>, <i>m</i>, <i>s</i>)-nets
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Bibliographic record
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}}\in\mathbb{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 <jats:inline-formula
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it