Statistical Consistency for Risk Measures With the Lebesgue Property
Bibliographic record
Abstract
<p>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].</p>
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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 | 0.003 | 0.002 |
| 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 | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".