When Heavy Tails Disrupt Statistical Inference
Bibliographic record
Abstract
Heavy tails (HT) arise in many applications and their presence can disrupt statistical inference, yet the HT statistical literature requires a theoretical background most practicing statisticians lack. We provide an overview of the influence of HT on the performance of basic statistical methods and useful theorems aimed at the practitioner encountering HT in an applied setting. Higher or even lower product moments (i.e., variance, skewness, etc.) can be infinite for some HT populations, yet all L-moments are always finite, given that the mean exists, thus, the theory of L-moments is uniquely suited to all HT distributions and data. We document how L-kurtosis, (a kurtosis measure based on the fourth L-moment) provides a general and practical heaviness index for contrasting tail heaviness across distributions and datasets and how a single L-moment diagram can document both the prevalence and impact of HT distributions and data across disciplines and datasets. Surprisingly, the theory of L-moments, an extension and evolution of probability weighted moments, has been largely overlooked by the literature on HT distributions that exhibit infinite moments. Experiments reveal L-kurtosis ranges under which various HT distributions result in mild to severe disruption to the bootstrap, the central limit theorem (CLT), and the law of large numbers, even for distributions which exhibit finite product moments.
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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 | 0.065 | 0.383 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.002 | 0.008 |
| Scholarly communication | 0.007 | 0.011 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.004 | 0.007 |
| 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".