Fitting the Modified-Power-Lognormal to the Sum of Independent Lognormals Distribution
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Bibliographic record
Abstract
We propose a new method for calculating a tight approximation to the distribution of the sum of independent lognormal random variables. We make use of a three-parameter modified-power-lognormal distribution function as the approximating distribution. We use theoretical results from our previous work on the tails of the distribution of the sum of lognormals to match the slope of the modified-power-lognormal function at both tails. This would not have been possible with many of the recently-proposed distribution functions, which do not behave properly in the tails. We then also use moment-matching to find the best curve match. Our method is mostly closed-form, requiring only one simple numerical integral. We compare our method with those in literature in terms of complexity and accuracy. We conclude that our method is more accurate than the simple (closed-form) methods, and much simpler to understand and implement than the more accurate methods which rely heavily on numerical integration.
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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 | 0.002 | 0.000 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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