Improved Confidence Intervals for Quantiles of the Gumbel Probability Distribution Used in Modeling Extreme Hydrological Events
Bibliographic record
Abstract
An important problem in hydrology is the frequency modeling of extreme events such as floods or heavy rainfall. The Gumbel model is a probability distribution often used to describe the behavior of such events. This distribution is a special case of the generalized extreme values model (GEV). A key goal in fitting frequency distributions to data is to allow the estimation of distribution quantiles, often used as “design events”. The maximum likelihood (ML) method is recommended for fitting the Gumbel model to data. One way of measuring the statistical error involved in estimating design events is to calculate confidence intervals for quantiles (CIQs). Hydrologists have traditionally used large-sample theory to construct such CIQs, but we show that this leads to inaccurate results for quantiles in the right tail of a Gumbel model, when the sample size is not sufficiently large. We therefore propose an improvement to the classically obtained CIQs by applying a Box-Cox (BC) power transformation to the quantiles. The conventional and proposed approaches are then compared through Monte Carlo simulation. Practical recommendations are put forward and demonstrated in a hydrological application.
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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 | 0.021 | 0.106 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.003 | 0.003 |
| 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".