Accurate approximation of the expected value, standard deviation, and probability density function of extreme order statistics from Gaussian samples
Why this work is in the frame
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Bibliographic record
Abstract
We show that the expected value of the largest order statistic in Gaussian samples can be accurately approximated as (0.2069 ln (ln (n))+0.942)4, where n∈[2,108] is the sample size, while the standard deviation of the largest order statistic can be approximated as −0.4205arctan(0.5556[ln(ln (n))−0.9148])+0.5675. We also provide an approximation of the probability density function of the largest order statistic which in turn can be used to approximate its higher order moments. The proposed approximations are computationally efficient, and improve previous approximations of the mean and standard deviation given by Chen and Tyler (1999 Chen, C.-C., and C. W. Tyler. 1999. Accurate approximation to the extreme order statistics of Gaussian samples. Communications in Statistics-Simulation and Computation 28 (1):177–88. doi:https://doi.org/10.1080/03610919908813542.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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