On consistency of the least squares estimators in linear errors-in-variables models with infinite variance errors
Bibliographic record
Abstract
This paper deals simultaneously with linear structural and functional errors-in-variables models (SEIVM and FEIVM), revisiting in this context the ordinary least squares estimators (LSE) for the slope and intercept of the corresponding simple linear regression. It has been known that, subject to some model conditions, these estimators become weakly and strongly consistent in the linear SEIVM and FEIVM with the measurement errors having finite variances when the explanatory variables have an infinite variance in the SEIVM, and a similar infinite spread in the FEIVM, while otherwise, the LSE’s require an adjustment for consistency with the so-called reliability ratio. In this paper, weak and strong consistency, with and without the possible rates of convergence being determined, is proved for the LSE’s of the slope and intecept, assuming that the measurement errors are in the domain of attraction of the normal law (DAN) and thus are, for the first time, allowed to have infinite variances. Moreover, these results are obtained under the conditions that the explanatory variables are in DAN, have an infinite variance, and dominate the measurement errors in terms of variation in the SEIVM, and under appropriately matching versions of these conditions in the FEIVM. This duality extends a previously known interplay between SEIVM’s and FEIVM’s.
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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 | 0.076 | 0.338 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.008 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.005 | 0.006 |
| Research integrity | 0.004 | 0.006 |
| Insufficient payload (model declined to judge) | 0.002 | 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".