Ordering results for the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables
Bibliographic record
Abstract
In this paper, we discuss stochastic comparisons of the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables. Let X1,…,Xn be independent random variables with Xi∼EW(αi,βi,γi), i=1,…,n. Further, let X1∗,…,Xn∗ be another set of independent random variables with Xi∗∼EW(αi∗,βi∗,γi∗), i=1,…,n. First, when γ1=⋯=γn=γ1∗=⋯=γn∗ and a matrix with different parameters αi,βi changes to another matrix in the sense of multivariate chain majorization and row majorization, we investigate the usual stochastic order of the largest order statistics. Next, when α1=⋯=αn=α1∗=⋯=αn∗,β1=⋯=βn=β1∗=⋯=βn∗ and (γ1,…,γn)⪰m(γ1∗,…,γn∗), we establish the usual stochastic order of the largest and smallest order statistics. Finally, we provide sufficient conditions for the hazard rate order of the smallest order statistics.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.011 | 0.054 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".