BAYES AND EMPIRICAL BAYES ESTIMATORS WITH THEIR UNIQUE SIMPLER FORMS AND THEIR SUPERIORITIES OVER BLUE IN TWO SEEMINGLY UNRELATED REGRESSIONS
Bibliographic record
Abstract
This paper considers Bayes and empirical Bayes estimation of the regression parameters in a system of two seemingly unrelated regression (SUR) models with Gaussian disturbances. Employing the covariance-adjusted technique, we obtain a sequence of Bayes estimators and show that this is the best Bayes estimator (BE) in the sense of having least covariance matrix. We establish the superiority of this best BE over the best linear unbiased estimator (BLUE) in terms of the mean square error matrix (MSEM) criterion. When the covariance matrix of disturbances is unknown, we obtain a sequence of corresponding empirical Bayes (EB) estimators and show that this too is superior to BLUE in MSEM criterion. In addition, we establish an interesting fact which shows that both Bayes and empirical Bayes estimators have only unique simpler forms, which too are superior to BLUE, and further allow us to study them in more details. This paper shows that the proposed Bayes and EB estimators, which combine the Bayesian method with the covariance-adjusted technique, are very efficient even for small sample size. Finally, we generalize our results to the system of two SURs with unequal numbers of observations and to the system of more than two SURs.
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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 | 0.028 | 0.127 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 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".