System Stress-Strength Reliability: The Multivariate Case
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Bibliographic record
Abstract
<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Present day complex systems with dependence between their components require more advanced models to evaluate their reliability. We compute the reliability of a system consisting of two subsystems <formula formulatype="inline"> <tex>$S_{1}$</tex></formula>, and <formula formulatype="inline"><tex>$S_{2}$</tex> </formula> connected in series, where the reliability of each subsystem is of general stress-strength type, defined by <formula formulatype="inline"> <tex>$\Re_{1}=P({\bf A}^{T}{\bf X}>{\bf B}^{T}{\bf Y})$</tex></formula>. <formula formulatype="inline"><tex>${\bf A}$</tex></formula> & <formula formulatype="inline"> <tex>${\bf B}$</tex></formula> are column-constant vectors, and strength <formula formulatype="inline"><tex>${\bf X}$</tex></formula> & stress <formula formulatype="inline"><tex>${\bf Y}$</tex></formula> are multigamma random vectors, i.e. <formula formulatype="inline"><tex>$({\bf X},{\bf Y})\sim MG({\mmb \alpha},{\mmb \beta})$</tex></formula>, where <formula formulatype="inline"> <tex>${\mmb \alpha}$</tex></formula> and <formula formulatype="inline"><tex>${\mmb \beta}$</tex></formula> are k-dimensional constant vectors. A Bayesian approach is adopted for <formula formulatype="inline"><tex>$\Re_{2}=P({\bf B}^{T}{\bf W}\geq 0)$</tex></formula>, where <formula formulatype="inline"><tex>${\bf W}$</tex></formula> is multinormal, i.e.<formula formulatype="inline"><tex>${\bf W}\sim MN({\mmb \mu},{\bf T})$</tex></formula>, with the mean vector <formula formulatype="inline"><tex>${\mmb \mu}$</tex></formula>, and the precision matrix <formula formulatype="inline"><tex>${\bf T}$</tex></formula> having a joint <formula formulatype="inline"><tex>$s$</tex></formula>-normal-Wishart prior distribution. Final computations are carried out by simulation, an approach which plays a major role in this article. The results obtained show that the approach adopted can deal effectively with the dependence between components of <formula formulatype="inline"><tex>${\bf X}$</tex></formula> & <formula formulatype="inline"><tex>${\bf Y}$</tex></formula>. </para>
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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