On Approximating the Distribution of the Durbin-Watson Statistic from its Moments Obtained Recursively
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Bibliographic record
Abstract
SYNOPTIC ABSTRACTA recursive relationship for determining the moments of a quadratic form in normal variables as well as an explicit formula for approximating a continuous density function defined on a compact support from its moments are derived in this paper. Each of these results have, on their own, a plethora of applications as quadratic forms are ubiquitous in Statistics and the moments of most test statistics that are confined to closed intervals can be readily evaluated; they are combined herewith to produce an approximation to the null distribution of the Durbin-Watson statistic, which for all intents and purposes, can be viewed as exact. The proposed approach takes into account the observation matrix of explanatory variables associated with the assumed regression model, and more accuracy can always be gained by making use of additional moments. Furthermore, the Durbin-Watson statistic is shown to be invariant in the class of spherically distributed error vectors, and an integral formula is derived for evaluating its moments under the assumption that the error vector has a general covariance structure. A numerical example illustrates the proposed methodology.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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