Bounds for Probabilities of Extreme Events Defined by Two Random Variables
Bibliographic record
Abstract
This paper offers a methodology for calculating optimal bounds on tail risk probabilities by deriving upper and lower semiparametric bounds, given only the first two moments of the distribution. We apply this methodology to determine bounds for probabilities of two tail events. The first tail event occurs when two financial variables simultaneously have extremely low values. The second occurs when the sum of two financial variables takes a very low value. In both cases we are finding bounds for actual or physical probabilities of these events rather than probabilities for a pricing or risk neutral measure. We use sum of squares optimization programs to obtain the desired bounds. To illustrate our ideas, we present several numerical examples. This approach is suitable in the situations when it is difficult to make exact distributional assumptions due to, for instance, scarcity and/or high volatility of data. Even in the situations when distributional assumptions can be made, this approach can be used to check the consistency of those assumptions.
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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 | 0.018 | 0.087 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".