Moment Problem and Its Application to Tail Risk Assessment
Bibliographic record
Abstract
The paper discusses how to assess risk by computing the best upper and lower bounds on the expected value E[φ(X)], subject to the constraints E[Xi] = µi for i = 0, 1, 2, . . . , n. φ(x) can take the form of the indicator function φ(x) = 𝕀(−∞,K](x) in which the bounds on Pr(X ≤ K) are calculated and the form φ(x) = (ϕ(x)−K)+ in which the bounds on financial payments are founds. We solve the moment bounds on E[𝕀(−∞,K](X)] through three methods; namely, the semidefinite programming method, the moment-matching method, and the linear approximation method. We show that for practical purposes, these methods provide numerically equivalent results. We explore the accuracy of bounds in terms of the number of moments considered. We investigate the usefulness of the moment method by comparing the moment bounds with the “point” estimate provided by the Johnson System of distributions. In addition, we propose a simpler formulation for the unimodal bounds on E[𝕀(−∞,K](X)] compared to the existing formulations in the literature. Furthermore, for those problems that could be solved both analytically and numerically given the first few moments, our comparisons between the numerical and analytical results call attention to the potential differences between these two methodologies. Our analysis indicates the numerical bounds could deviate from their corresponding analytical counterparts. The accuracy of numerical bounds is sensitive to the volatility of X. The more volatile the random variable X is, the looser the numerical bounds are compared to their closed-form solutions.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.006 | 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 | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".