Maximum L<sub><i>q</i></sub>-likelihood Estimation for Gamma Distributions
Bibliographic record
Abstract
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model.Standard large sample theory guarantees asymptotic efficiency of MLE.On the other hand, MLE does not perform as well as expected for moderate or small sample size.In 2010, a new parameter estimator based on nonextensive entropy ([1]), named Maximum Lq-likelihood Estimator (MLqE), was first introduced and studied by [2].MLqE is an extension of MLE which introduces a distortion parameter q to make the estimation more adaptive.The purpose of this work is to examine this methodology for gamma distributions that are widely used in engineering, science and business to model continuous but skewed distributions.For specifically standard gamma models, we look at the MLqE's asymptotics, finite sample performance in terms of efficiency and robustness, and the choice of the distortion parameter q.We investigate these aspects of MLqE and compare it with MLE in parameter estimation and tail probability estimation, through both Monte Carlo simulation and a real data analysis.Our results show that, with appropriately chosen q, MLqE and MLE perform competitively for large sample sizes while MLqE outperforms MLE for small or moderate sample sizes in terms of reducing MSE.In addition, MLqE with q < 1 has much better robustness properties than MLE when outlying observations are present.
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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 | 0.004 | 0.029 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.003 |
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".