Erdelyi-Kober Fractional Integral Operators from a Statistical Perspective -II
Bibliographic record
Abstract
In this article we examine the densities of a product and a ratio of two real positive definite matrix-variate random variables $X_1$ and $X_2$, which are statistically independently distributed, and we consider the density of the product $U_1=X_2^{1\over2}X_1X_2^{1\over2}$ as well as the density of the ratio $U_2=X_2^{1\over2}X_1^{-1}X_2^{1\over2}$. We define matrix-variate Kober fractional integral operators of the first and second kinds from a statistical perspective, making use of the derivation in the predecessor of this paper for the scalar variable case, by deriving the densities of products and ratios where one variable has a matrix-variate type-1 beta density and the other variable has an arbitrary density. Various types of generalizations are considered, by using pathway models, by appending matrix variate hypergeometric series etc. During this process matrix-variate Saigo operator and other operators are also defined and properties studied.
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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 | 0.004 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
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".