Estimation of P(Y<X) for a Two-parameter Bathtub Shaped Failure Rate Distribution
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Bibliographic record
Abstract
This paper deals with the estimation of reliability R = P[Y < X] when X and Y are two independent random variables with atwo-parameter bathtub shaped failure rate distribution with the samesecond shape parameter. Likelihood and Bayesian methods are proposedto make inferences about R. We obtain the likelihood interval andasymptotic confidence interval for R, and we consider Bayesianpoint estimates of R under both absolute and squared error loss,using either gamma or uniform priors for the three unknown modelparameters. An equal tail Bayesian credible interval for R isinvestigated. Analysis of a real data set is presented forillustrative purposes, and Monte Carlo simulations are performed tocompare: (1) the performance of Bayes estimates under two differentloss functions; and (2) the maximum likelihood and Bayesian methods.
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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 | 0.009 |
| 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