On a Family of Weighted Cramér–Von Mises Goodness-of-Fit Tests in Operational Risk Modeling
Bibliographic record
Abstract
The measurement of operational risk via the loss distribution approach (LDA) for bank capitalization purposes offers significant modeling challenges. Under the LDA, the severity of losses characterizing the monetary impact of potential operational risk events is modeled via a severity distribution. The selection of best-fit severity distributions that properly capture tail behavior is essential for accurate modeling. In this paper, we analyze the limiting properties of a family of weighted Cramér–von Mises (WCvM) goodness-of-fit test statistics, with weight function ψ⠢(t)=1/(1-t)β, which are suitable for more accurately selecting severity distributions. Specifically, we apply classical theory to determine if limiting distributions exist for these WCvM test statistics under a simple null hypothesis. We show that limiting distributions do not exist for β≥2. For β=2, we provide a normalization that leads to a nondegenerate limiting distribution. Where limiting distributions originally exist for β<2 or are obtained through normalization, we show that, for 1.5≤β≤2, the tests’ practical utility may be limited due to a very slow convergence of the finite-sample distribution to the asymptotic regime. Our results suggest that the tests provide greater utility when β<1.5, and that utility is questionable for β≥1.5, as only Monte Carlo schemes are practical in this case, even for very large samples.
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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 | 0.038 | 0.185 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.006 | 0.004 |
| Science and technology studies | 0.001 | 0.011 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.003 | 0.005 |
| 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".