Log-Shifted Gamma Approximation to Lognormal Sum Distributions
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Bibliographic record
Abstract
This paper proposes the log-shifted gamma (LSG) approximation to model the sum of M lognormally distributed random variables (RVs). The closed-form probability density function of the resulting LSG RV is presented, and its parameters are directly derived from those of the M individual lognormal RVs by using an iterative moment-matching technique without the need for curve fitting of computer-generated distributions. Simulation and analytical results on the cumulative distribution function (cdf) of the sum of M lognormal RVs in different conditions indicate that the proposed LSG approximation can provide better accuracy than other lognormal approximations over a wide cdf range, especially for large M and/or standard deviation.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| 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