Models with a Kronecker product covariance structure: Estimation and testing
Bibliographic record
Abstract
In this article we consider a pq-dimensional random vector x distributed normally with mean vector θ and covariance matrix Λ assumed to be positive definite. On the basis of N independent observations on the random vector x, we want to estimate parameters and test the hypothesis H: Λ = Ψ ⊗ Σ, where Ψ = (ψ ij ): q × q, ψ qq = 1, and Σ = (σ ij ): p × p, and Λ = (ψ ij Σ), the Kronecker product of Ψ and Σ. That is instead of 1/2pq(pq + 1) parameters, it has only 1/2p(p + 1) + 1/2q(q + 1) − 1 parameters. A test based on the likelihood ratio is given to check if this model holds. And, when this model holds, we test the hypothesis that Ψ is a matrix with intraclass correlation structure. The maximum likelihood estimators (MLE) are obtained under the hypothesis as well as under the alternatives. Using these estimators the likelihood ratio tests (LRT) are obtained. One of the main objects of the paper is to show that the likelihood equations provide unique estimators.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.038 | 0.175 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.009 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.004 | 0.004 |
| 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".