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Low Moments for Small Samples: A Comparative Study of Order Statistics
Open accessAbstract
The means, variances, and covariances for samples of size $\leq 10$ from the normal distribution, a selected long-tailed distribution, and the uniform distribution are tabled and compared with the usual asymptotic approximations. The methods of computation used and the accuracy expected are discussed. Use is made of the representation of an arbitrarily distributed variate as a monotone function of a uniformly (rectangularly) distributed variate. It is hoped that these tables will encourage experimentation with new statistical procedures.
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“For example, outliers in linear models may strongly impact linear estimators, and some studies may therefore adopt a modeling strategy wherein values are winsorized (e.g., extreme values can be truncated at the 2.5th and 97.5th quantiles56).”
From Guidelines for wrist-worn consumer wearable assessment of heart rate in biobehavioral research (2020) · cites this paper as [56]
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Metadata
- Authors
- Hastings, C., Mosteller, F., Tukey, J. W., & Winsor, C. P.
- Venue
- The Annals of Mathematical Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1947-09
- Volume
- 18
- Issue
- 3
- Pages
- 413-426
- ISSN
- 0003-4851 (ISSN-L)
- OpenAlex
- W2040509998
- Cited by
- 366
- References
- 0
- Citations by year
- 2026: 18 · 2025: 10 · 2024: 10 · 2023: 14 · 2022: 18 · 2021: 27 · 2020: 24 · 2019: 22 · 2018: 18 · 2017: 26 · 2016: 17 · 2015: 17 · 2014: 14 · 2013: 14 · 2012: 14
- FWCI
- 6.85
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Mathematical Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Keywords
- order statistics, moments, normal distribution, long-tailed distributions, uniform distribution, asymptotic approximation, mathematical tables
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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