Modelling N- and W-shaped hazard rate functions without mixing distributions
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Bibliographic record
Abstract
The presence of non-conforming components instead of, or in addition to, the usual assembly errors results in N- or W-shaped hazard rate (HR) functions rather than the usual bathtub (i.e. U-shaped) ones. Although there have been numerous models for bathtub-shaped HR functions, N- and W-shaped HR functions are usually modelled using mixtures of two or more distributions. While this approach does sometimes lead to tidy interpretation, there can be a degree of overparameterization, with consequent problems in stability and fitting. For this reason, the present paper revisits the natural approach of modelling N- and W-shaped HR functions using polynomial functions of degree three or four. Although the non-negativity of the hazard rate function becomes non-trivial, this ensures a minimal number of parameters. The polynomial approach also allows the use of a parametric model without imposing a particular shape of hazard rate function on the data, which usually requires a non-parametric approach. The possible hazard rate shapes obtainable are characterized, and detailed formulae for local minima and maxima of the functions provided. The performance of the models is compared to that of several generalizations of the Weibull distribution, with promising results.
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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 | 0.003 |
| 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