Likelihood ratio tests with boundary constraints using data-dependent degrees of freedom
Bibliographic record
Abstract
When the null hypothesis constrains parameters to the boundary of the parameter space, the asymptotic null distribution of the likelihood ratio statistic is often a mixture of chi-squared distributions, giving rise to the so-called chi-bar test, where weights can depend on the true unknown parameter and be difficult to calculate. We consider the test that conditions on the observed number of null hypothesis parameters in the interior of the parameter space. This approach uses simple chi-squared thresholds, yields conservative asymptotic Type I error, and is guaranteed to give power improvements over the naive approach of simply ignoring boundary constraints. Simulations validate the theoretical results, illustrate application settings, and find power comparable with but usually less than that of the chi-bar test.
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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 | 0.082 | 0.478 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.006 | 0.003 |
| Bibliometrics | 0.004 | 0.005 |
| Science and technology studies | 0.002 | 0.010 |
| Scholarly communication | 0.006 | 0.014 |
| Open science | 0.007 | 0.010 |
| Research integrity | 0.008 | 0.010 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".