Some tests criteria for the covariance matrix with fewer observations than the dimension
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Bibliographic record
Abstract
We consider testing certain hypotheses concerning the covariance matrix Σ when the number of observations N=n+1 on the p-dimensional random vector x, distributed as normal, is less than p, n<p, and n/p goes to zero. Specifically, we consider testing Σ=σ2Ip, Σ=Ip, Σ=Λ, a diagonal matrix, and Σ=σ2[(1−ρ)Ip+ρ1p1′p], an intraclass correlation structure, where 1′p=(1,1,…,1), is a p-row vector of ones, and Ip is the p×p identity matrix. The first two tests are the adapted versions of the likelihood ratio tests when n>p, p-fixed, and p/n goes to zero, to the case when n<p, n-fixed, and n/p goes to zero. The third test is the normalized version of Fisher’s z-transformation which is shown to be asymptotically normally distributed as n and p go to infinity (irrespective of the manner). A test for the fourth hypothesis is constructed using the spherecity test for a (p−1)-dimensional vector but this test can only reject the hypothesis, that is, if the hypothesis is not rejected, it may not imply that the hypothesis is true. The first three tests are compared with some recently proposed tests.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| 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 |
Machine scores (provisional)
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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 it