Extreme-Case Distortion Risk Measures: A Unification and Generalization of Closed-Form Solutions
Why this work is in the frame
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Bibliographic record
Abstract
Extreme-case risk measures provide an approach for quantifying the upper and lower bounds of risk in situations where limited information is available regarding the underlying distributions. Previous research has demonstrated that for popular risk measures, such as value-at-risk and conditional value-at-risk, the worst-case counterparts can be evaluated in closed form when only the first two moments of the underlying distributions are known. In this study, we extend these findings by presenting closed-form solutions for a general class of distortion risk measures, which consists of various popular risk measures as special cases when the first and certain higher-order (i.e., second or more) absolute center moments, alongside the symmetry properties of the underlying distributions, are known. Moreover, we characterize the extreme-case distributions with convex or concave envelopes of the corresponding distributions. By providing closed-form solutions for extreme-case distortion risk measures and characterizations for the corresponding distributions, our research contributes to the understanding and application of risk quantification methodologies. Funding: H. Shao acknowledges support from the Yangtze River Delta Science and Technology Innovation Community Joint Research Program [Grant 2022CSJGG0800]. Z. G. Zhang acknowledges support from the Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2019-06364]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2022.0156 .
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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 | 0.009 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| 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)
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