Asymptotic Confidence Ellipses of Parameters for the Beta-Poisson Dose-Response Model
Bibliographic record
Abstract
The objective of this study is to argue that two classical estimation methods (method of moments and method of maximum likelihood) cannot be applied for a construction of an asymptotic confidence ellipse for the Beta-Poisson dose-response model directly. That is, the first moment for the probability density function of the Beta-Poisson dose-response model exists for > 1 , but data known from literature show that a typical values of is around a few hundredths, while a typical values of is around a few hundred. And the method of maximum likelihood, we cannot find parameter estimators of the Beta-Poisson dose-response model and also it is not possible to find the asymptotic covariance matrix from the Fisher information matrix, hence it is also not possible to construct an asymptotic confidence ellipse based on the maximum likelihood estimates. For this purpose we need to use a suitable approximation for the Beta-Poisson dose-response model. After that, we derive maximum likelihood equations for parameters of the approximate Beta-Poisson dose-response model. For this model, we find the Fisher information matrix and construct a normal approximation that gives elliptical confidence regions of the approximate Beta-Poisson dose-response model. We use the coverage probabilities as the criteria for comparing with the nominal level 0.98 and investigate the accuracy of the confidence ellipses are fulfilled by the Monte-Carlo method. Three sample sizes (n) are 100, 500, 1,000 and four cases of values for parameters of the Beta-Poisson dose-response model ((i) = 0.1, = 10 (ii) = 0.08, = 100 (iii) = 0.3, = 150 (iv) = 0.7, = 180 ) are studied here. R (2.13.0) software is used for simulation technique with 10,000 iterations. The results of this study are as followed: The coverage probabilities of confidence ellipses for parameters of the approximate Beta-Poisson dose-response model increase when sample sizes (n) increase and also they are close to the confidence coefficient 0.98. In addition, at various values of parameters of Beta-Poisson dose-response model, the coverage probabilities are similar when n is fixed.
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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 | 0.036 | 0.192 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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".