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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Outliers in linear models may strongly impact linear estimators, and some studies may therefore adopt a modeling strategy wherein values are winsorized.
“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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“This outlier was replaced by the variable mean plus three times its standard deviation (Hasings et al., 1947).”
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“This is similar to winsorising that sets the extreme probabilities to a specified percentile of the data [see, e.g., Hastings et al. (1947) for more details on winsorising].”
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“Although trimming 1 % of the sample is one possibility, a more appropriate approach for this analysis is to Winsor the data, thereby retaining the small sample size for the D/S category and maintaining statistical power (Hastings et al. 1947; Tukey 1962; Salkind 2010).”
MethodA Uniformly Asymptotically Efficient Estimator of a Location Parameter
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Evaluating English language learners’ conversations: Man vs. Machine
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“Outliers within the data were replaced via the Winsor method (Hastings, Mosteller, Tukey, & Winsor, 1947).”
MethodYu, C., Zou, Z., Shao, Y., et al. (2019) · Kybernetes · doi:10.1108/k-04-2019-0223 · Cited by 30 · see citing works
Murea, M., Register, T. C., Divers, J., Bowden, D. W., Carr, J. J., Hightower, C. R., Xu, J., Smith, S. C., Hruska, K. A., Langefeld, C. D., et al. (2012) · BMC Nephrology · doi:10.1186/1471-2369-13-148 · Cited by 28 · see citing works
“7 [28].”
Learner Groups in Massive Open Online Courses
Arora, S., Goel, M., Sabitha, A. S., et al. (2017) · American Journal of Distance Education · doi:10.1080/08923647.2017.1300461 · Cited by 28 · see citing works
“This technique of outlier treatment is called Winsorization or Percentile Capping — substituting extreme values with less extreme values (Hastings et al. 1947).”
MethodPoltoratski, S., & Tong, F. (2020) · Journal of Neuroscience · doi:10.1523/jneurosci.2377-19.2020 · Cited by 26 · see citing works
“Outliers were defined as time points for which the voxel’s response measured more than 3 times its standard deviation from its mean, and were Winsorised (Hastings et al., 1947).”
BackgroundGalbraith, A. A., Meyers, D. J., Ross‐Degnan, D., Burns, M. E., Vialle-Valentin, C. E., Larochelle, M. R., Touw, S., Zhang, F., Rosenthal, M. B., & Balaban, R. B. (2017) · Health Services Research · doi:10.1111/1475-6773.12790 · Cited by 27 · see citing works
Empirical distributions of daily equity index returns: A comparison
Corlu, C. G., Meterelliyoz, M., & Tiniç, M. (2016) · Expert Systems with Applications · doi:10.1016/j.eswa.2015.12.048 · Cited by 26 · see citing works
Kafadar, K. (2003) · Statistical Science · doi:10.1214/ss/1076102419 · Cited by 26 · see citing works
Extended Models for Quantal Response Data
Morgan, B. J. T. (1988) · Statistica Neerlandica · doi:10.1111/j.1467-9574.1988.tb01239.x · Cited by 26 · see citing works
Mehrotra, R., Stanaway, I. B., Jarvik, G. P., Lambie, M., Morelle, J., Perl, J. M., Himmelfarb, J., Heimbürger, O., Johnson, D. W., Imam, T. H., et al. (2021) · Kidney International · doi:10.1016/j.kint.2021.05.037 · Cited by 25 · see citing works
A Discretized Approach to Flexibly Fit Generalized Lambda Distributions to Data
Su, S. Y. S. (2005) · Journal of Modern Applied Statistical Methods · doi:10.22237/jmasm/1130803560 · Cited by 25 · see citing works
ciRS-7 and miR-7 regulate ischemia-induced neuronal death via glutamatergic signaling
Scoyni, F., Sitnikova, V., Giudice, L., Korhonen, P., Trevisan, D. M., Hernandez de Sande, A., Gomez-Budia, M., Giniatullina, R., Ugidos, I. F., Dhungana, H., et al. (2024) · Cell Reports · doi:10.1016/j.celrep.2024.113862 · Cited by 28 · see citing works
Deimling, G. T., Pappada, H. T. R., Ye, M., Nalepa, E., Ciaralli, S. R., Phelps, E. K., & Burant, C. J. (2017) · Journal of Aging and Health · doi:10.1177/0898264317745745 · Cited by 24 · see citing works
Poorer sleep impairs brain health at midlife
Namsrai, T., Ambikairajah, A., & Cherbuin, N. (2023) · Scientific Reports · doi:10.1038/s41598-023-27913-9 · Cited by 25 · see citing works
- 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
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