GENERATING FUNCTIONS AND SHORT RECURSIONS, WITH APPLICATIONS TO THE MOMENTS OF QUADRATIC FORMS IN NONCENTRAL NORMAL VECTORS
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Bibliographic record
Abstract
Recursive relations for objects of statistical interest have long been important for computation, and they remain so even with hugely improved computing power. Such recursions are frequently derived by exploiting relations between generating functions. For example, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other (easily computed) symmetric functions (power-sum and elementary symmetric functions; Ruben, 1962, Annals of Mathematical Statistics 33, 542–570; Hillier, Kan, and Wang, 2009, Econometric Theory 25, 211–242). Typically, in a recursion of this type the k th object of interest, d k , say, is expressed in terms of all lower order d j ’s. In Hillier et al. (2009) we pointed out that, in the case of top-order zonal polynomials and other invariant polynomials of multiple matrix argument, a fixed length recursion can be deduced. We refer to this as a short recursion. The present paper shows that the main results in Hillier et al. (2009) can be generalized and that short recursions can be obtained for a much larger class of objects/generating functions. As applications, we show that short recursions can be obtained for various problems involving quadratic forms in noncentral normal vectors, including moments, product moments, and expectations of ratios of powers of quadratic forms. For this class of problems, we also show that the length of the recursion can be further reduced by an application of a generalization of Horner’s method (cf. Brown, 1986, SIAM Journal on Scientific and Statistical Computing 7, 689–695), producing a super-short recursion that is significantly more efficient than even the short recursion.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 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)
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 it