Maximum Entropy Estimation of Density Function Using Order Statistics
Bibliographic record
Abstract
The main premise of this article is to develop a maximum entropy estimation of an unknown distribution using order statistics. The exact solution using constraints on the order statistic means is derived. The result is an integral of a rational parametric function whose parameters depend on the order statistic constraints. This integral equation can be solved directly for a small number of order statistic means, e.g., 1, 2, or at most 3, via a multidimensional search in the parameter space. As the number of constraints increases, the search for the optimum parameters becomes formidable. To resolve this problem we have proposed an approximate solution to the resultant integral equation based on Bernstein polynomials and estimates of a desired number of quantiles from available samples. The proposed approximation approach does not require any search in the parameter space and is formulated for any number of samples from an unknown distribution. Performance of this approach is evaluated under various practical conditions by using Kullback-Leibler divergence function.
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".