Statistical Consistency for Risk Measures With the Lebesgue Property
Bibliographic record
Abstract
When estimating the risk ρ(X) of a random variable X from historical data or Monte Carlo simulation, the asymptotic behaviour of the plug in estimator ρbn is of utmost importance. In their celebrated article [19], the Kra ̈atschmer et al. showed that any finite-valued law-invariant convex risk measure ρ defined on an Orlicz heart HΦ is statistically consistent. That is, the plug-in estimator ρbn converges in the almost sure sense to ρ(X). This result is very general, yet it does not cover the case where ρ is non-convex nor the case where ρ is defined on the entire Orlicz space L Φ. The aim of this thesis is to fill this gap. In particular, we prove that any law-invariant risk measure with the Lebesgue property is statistically consistent on the entire Orlicz space. The Lebesgue property is a continuity condition that is automatically satisfied by all convex and finite-valued risk measures on Orlicz hearts. Thus our result can be viewed as a generalization of Theorem 2.6 in [19].
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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 | 0.019 | 0.076 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 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".