Estimating Dispersion Parameter of Negative Binomial Distribution for Analysis of Crash Data
Bibliographic record
Abstract
The objective of this study is to improve the estimation of the dispersion parameter of the negative binomial distribution for modeling motor vehicle collisions. The negative binomial distribution is widely used to model count data such as traffic crash data, which often exhibit low sample mean values and small sample sizes. Under such situations, the most commonly used methods for estimating the dispersion parameter–the method of moment and the maximum likelihood estimate–may become inaccurate and unstable. A bootstrapped maximum likelihood method is proposed to improve the estimation of the dispersion parameter. The proposed method combines the technique of bootstrap resampling with the maximum likelihood estimation method to obtain better estimates of the dispersion parameter. The performance of the bootstrapped maximum likelihood estimate is compared with the method of moment and the maximum likelihood estimates through Monte Carlo simulations. To validate the simulation results, the methods are applied to observed data collected at four-leg unsignalized intersections in Toronto, Ontario, Canada. Overall, the results show that the proposed bootstrap maximum likelihood method produces smaller biases and more stable estimates. The improvements are more pronounced with small samples and low sample means.
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How this classification was reachedexpand
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 | 0.006 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".