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When Heavy Tails Disrupt Statistical Inference
Publikationstyp
Journal Article
Date Issued
2025
Sprache
English
Author(s)
Journal
American Statistician
Volume
79
Issue
2
Start Page
221
End Page
235
Citation
American Statistician 79 (2): 221-235 (2025)
Publisher DOI
Scopus ID
ISSN
00031305
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.
Subjects
Bootstrap
Central limit theorem
Fat tails
L-kurtosis
L-moment
Power-law
Product moment
Sample mean
Subexponential
DDC Class
620: Engineering