Orthogonal polynomials in the cumulative Ord family and its application to variance bounds
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Bibliographic record
Abstract
This article presents and reviews several basic properties of the Cumulative Ord family of distributions; this family contains all the commonly used discrete distributions. A complete classification of the Ord family of probability mass functions is related to the orthogonality of the corresponding Rodrigues polynomials. Also, for any random variable X of this family and for any suitable function g in L2(R,X), the article provides useful relationships between the Fourier coefficients of g (with respect to the orthonormal polynomial system associated to X) and the Fourier coefficients of the forward difference of g (with respect to another system of polynomials, orthonormal with respect to another distribution of the system). Finally, using these properties, a class of bounds for the variance of g(X) is obtained, in terms of the forward differences of g. These bounds unify and improve several existing results.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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