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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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Rendina, D. N., Blohowiak, S. E., Coe, C. L., & Kling, P. J. (2018) · The Journal of Pediatrics · doi:10.1016/j.jpeds.2018.04.040 · Cited by 25
“To limit the influence of extreme outliers, 1 RE-ZnPP/H and D-ZnPP/H value was winsorized and set at the upper 3 SD point.(35) ANCOVA and linear regression were used to test the association between maternal stress and the infants’ iron status.”
Stratton, N., Mendoza-Alvarez, M., Labrish, C., Barnhart, R., & McMain, S. (2018) · Journal of Personality Disorders · doi:10.1521/pedi_2018_32_391 · Cited by 28
Local Food Market Orientation and Labor Intensity
Jablonski, B. B. R., Bauman, A., & Thilmany, D. (2020) · Applied Economic Perspectives and Policy · doi:10.1002/aepp.13059 · Cited by 23
A conversation with John W. Tukey and Elizabeth Tukey
Fernholz, L. T., & Morgenthaler, S. (2000) · Statistical Science · doi:10.1214/ss/1009212675 · Cited by 23
Flexible distribution modeling with the generalized lambda distribution
Chalabi, Y., Scott, D. J., & Würtz, D. (2012) · Munich Personal RePEc Archive (Ludwig Maximilian University of Munich) · Cited by 23
ELKI: A large open-source library for data analysis - ELKI Release 0.7.5 "Heidelberg"Preprint
Schubert, E., & Zimek, A. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1902.03616 · Cited by 55
“bs.elki.math.statistics.distribution.estimator.meta.TrimmedEstimator Trimmed wrapper around other estimators. Sorts the data, trims it, then analyzes it using another estimator. Winsorizing Estimator [HMTW47] (since 0.6.0) de.lmu.ifi.dbs.elki.math.statistics.distribution.estimator.meta.WinsorizingEstimator Winsorizing…”
MethodInfluentialPapadakos, J., Barnsley, J. M., Berta, W. B., Rowlands, G., Samoil, D., & Howell, D. (2021) · Supportive Care in Cancer · doi:10.1007/s00520-021-06466-5 · Cited by 26
Jablonski, B. B. R., Hadrich, J. C., Bauman, A., et al. (2022) · Agricultural Finance Review · doi:10.1108/afr-05-2021-0056 · Cited by 22
Distributional Effects of Subsidizing Retirement Savings Accounts: Evidence from Germany
Corneo, G. G., & Schröder, C. (2018) · FinanzArchiv Public Finance Analysis · doi:10.1628/fa-2018-0017 · Cited by 22
“See Hastings Jr. et al. (1947).”
MethodExtended and multivariate Tukey lambda distributions
Johnson, N. L., & Kotz, S. (1973) · Biometrika · doi:10.1093/biomet/60.3.655 · Cited by 22
“INTRODUCTION The Tukey lambda distributions [2,7,8] have been found to be useful in a number of univariate sampling problems.”
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Jones, S. J., Draper, C. L., Bell, B. A., Burke, M. P., Martini, L., Younginer, N., Blake, C. E., Probst, J., Freedman, D. A., & Liese, A. D. (2018) · Journal of Hunger & Environmental Nutrition · doi:10.1080/19320248.2017.1364189 · Cited by 22
“Monthly wages and other monthly assistance were Winsorized (top and bottom coded) at the 95th percentile to reduce the influence of outliers.(42) Monthly income variables were divided by $100 so that the odds ratios (ORs) represent the increased or decreased odds of a household experiencing food insecurity or child hu…”
BackgroundPower and robustness of jackknife and likelihood-ratio tests for grouped heteroscedasticity
Sharma, S. C., & Giaccotto, C. (1991) · Journal of Econometrics · doi:10.1016/0304-4076(91)90002-u · Cited by 22
Roberts, S. E., Rosen, C. B., Keele, L. J., Wirtalla, C. J., Syvyk, S., Kaufman, E. J., Reilly, P. M., Neuman, M. D., McHugh, M. D., & Kelz, R. R. (2022) · JAMA Surgery · doi:10.1001/jamasurg.2022.4959 · Cited by 21
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Graff, R. E., Ahearn, T. U., Pettersson, A., et al. (2017) · Cancer Epidemiology Biomarkers & Prevention · doi:10.1158/1055-9965.epi-17-0547 · Cited by 21
“We reduced outlier influence on continuous exposures by employing a 99% Winsorization technique (28).”
MethodLeopold, C., Wagner, A. K., Zhang, F., Lu, C. Y., Earle, C. C., Nekhlyudov, L., Ross‐Degnan, D., & Wharam, J. F. (2018) · Breast Cancer Research and Treatment · doi:10.1007/s10549-018-4819-6 · Cited by 21
“1% (all numbers above zeros) of all health service spending [19].”
Dynamics of mutual funds and stock markets in Asian developing economies
Qureshi, F., Kutan, A. M., Ghafoor, A., et al. (2019) · Journal of Asian Economics · doi:10.1016/j.asieco.2019.101135 · Cited by 21
Hammer, R., Tennekoon, M., Cooke, G. E., Gayda, J., Stein, M. A., & Booth, J. R. (2015) · Developmental Cognitive Neuroscience · doi:10.1016/j.dcn.2015.06.002 · Cited by 21
“…an indication for reater engagement to the VSWM tasks, and thus we rescaled the eta values in this fROI using the formula ′ i = xi − max(x) min(x) − max(x) Prior to rescaling the Beta values, outlier values were replaced sing the Winsorising procedure (Hastings et al., 1947; Wilbertz t al., 2013).”
Estimating the Parameters of the Generalized Lambda Distribution: Which Method Performs Best?
Corlu, C. G., & Meterelliyoz, M. (2015) · Communications in Statistics - Simulation and Computation · doi:10.1080/03610918.2014.901355 · Cited by 21
“The generalized lambda distribution (Filliben, 1975; Joiner and Rosenblatt, 1971; Ramberg and Schmeiser, 1974), which is an extension of Tukey’s lambda distribution (Hastings et al., 1947) is defined by the following inverse cumulative distribution function (cdf):”
BackgroundIndividual, Family, and Peer predictors of Violence Among Samoan Adolescents
Stevens, D. L., & Hardy, S. A. (2011) · Youth & Society · doi:10.1177/0044118x11424756 · Cited by 24
Activating Lay Health Influencers to Promote Tobacco Cessation
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Malejka, S., & Bröder, A. (2018) · Journal of Memory and Language · doi:10.1016/j.jml.2018.09.001 · Cited by 18
Gilchrist, W. G. (2008) · International Statistical Review · doi:10.1111/j.1751-5823.2008.00053.x · Cited by 18
Opportunities to support beginning farmers and ranchers in the 2023 Farm Bill
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On the efficiencies of some common quick estimators
Mudholkar, G. S., Preimer, M., & Hutson, A. D. (1997) · Communication in Statistics- Theory and Methods · doi:10.1080/03610929708832005 · Cited by 18
Sihombing, P. R., Suryadiningrat, S., Sunarjo, D. A., et al. (2023) · Jurnal Ekonomi dan Statistik Indonesia · doi:10.11594/jesi.02.03.07 · Cited by 16
Willink, R. (2009) · Metrologia · doi:10.1088/0026-1394/46/3/002 · Cited by 16
Effects of Race and Language on Patient-Centered Cancer Nursing Care and Patient Outcomes
Radwin, L. E., Cabral, H., & Woodworth, T. S. (2013) · Journal of Health Care for the Poor and Underserved · doi:10.1353/hpu.2013.0058 · Cited by 16
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Satopää, V., Jensen, S. T., Pemantle, R., & Ungar, L. (2017) · Electronic Journal of Statistics · doi:10.1214/17-ejs1346 · Cited by 12
“Only the effect of extreme values was reduced via a 90% Winsorization (Hastings et al., 1947).”
BackgroundValue relevance of deferred tax assets in the Brazilian banking industry
Guia, L. D., & Dantas, J. A. (2019) · Revista Contabilidade & Finanças · doi:10.1590/1808-057x201808060 · Cited by 16
Awika, H. O., Cochran, K., Joshi, V., Bedre, R. H., Mandadi, K. K., & Avila, C. A. (2019) · Plant Breeding · doi:10.1111/pbr.12773 · Cited by 14
Gipson, D. S., Trachtman, H., Waldo, A., et al. (2019) · Kidney International Reports · doi:10.1016/j.ekir.2019.11.018 · Cited by 15
“73 m(2).(22) Urine protein and creatinine were measured in the central laboratory from 24-hour or spot urine samples as available.”
MethodExtremes,extreme spacings and outliers in the tukey and weibull families
Freimer, M. L., Kollia, G. D., Mudholkar, G. S., et al. (1989) · Communication in Statistics- Theory and Methods · doi:10.1080/03610928908830153 · Cited by 15
Pride-asap: Automatic fragment ion annotation of identified PRIDE spectra
Hulstaert, N., Reisinger, F., Rameseder, J., et al. (2013) · Journal of Proteomics · doi:10.1016/j.jprot.2013.04.011 · Cited by 14
“An adaptive noise filter based upon iterative winsorization [13] is first applied to each spectrum.”
BackgroundMethodThe Galaxy Stellar Mass Function and Low Surface Brightness Galaxies from Core-Collapse Supernovae
Sedgwick, T. M., Baldry, I. K., James, P. A., & Kelvin, L. S. (2019) · Monthly Notices of the Royal Astronomical Society · doi:10.1093/mnras/stz186 · Cited by 14
“(ii) Winsorize Petrosian Radii:Winsorization is the limiting of extreme values to reduce the effects of potentially spurious outliers (Hastings et al. 1947).”
BackgroundAn assumption-free exact test for fixed-design linear models with exchangeable errors
Lei, L., & Bickel, P. J. (2020) · Biometrika · doi:10.1093/biomet/asaa079 · Cited by 14
A Unified Approach to Robust Mean EstimationPreprint
Prasad, A., Balakrishnan, S., & Ravikumar, P. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1907.00927 · Cited by 14
“However, most of the estimators come that come with strong guarantees are computationally intractable [12], while others are statistically suboptimal heuristics [13].”
BackgroundOn the Regression Model for Generalized Normal Distributions
Alzaatreh, A., Aljarrah, M., Almagambetova, A., et al. (2021) · Entropy · doi:10.3390/e23020173 · Cited by 13
“After that, other methods were developed such as the method of transformation [3] and the method of quantile function [4,5].”
MethodKordovski, V. M., Tierney, S. M., Rahman, S., Medina, L. D., Babicz, M. A., Yoshida, H., Holcomb, E. M., Cushman, C., & Woods, S. P. (2021) · Journal of Clinical and Experimental Neuropsychology · doi:10.1080/13803395.2021.1990866 · Cited by 13
“Data for all tests with outliers underwent 95% Winsorization (Hastings et al., 1947).”
MethodGetting a Handle on Sales: Shopping Carts Affect Purchasing by Activating Arm Muscles
Estes, Z., & Streicher, M. C. (2021) · Journal of Marketing · doi:10.1177/00222429211061367 · Cited by 14
“To minimize alteration of the data, we replace those outliers with the most extreme nonoutlier among the observed data (i.e., Winsorizing; Hastings et al. 1947).”
MethodCorrective commit probability: a measure of the effort invested in bug fixing
Amit, I., & Feitelson, D. G. (2021) · Software Quality Journal · doi:10.1007/s11219-021-09564-z · Cited by 17
“In order to reduce the destabilizing effect of outliers, we applied Winsorizing (Hastings et al., 1947).”
MethodInfluentialCortisol responses to social evaluation in 10- to 15-year-old boys and girls
Zijlmans, M. A. C., Beijers, R., Mack, S., Pruessner, J. C., & de Weerth, C. (2013) · Stress · doi:10.3109/10253890.2013.764494 · Cited by 13
“All outliers were replaced by the group mean plus three standard deviations (Hasings et al. 1947).”
Reconciliation of outliers in CO2-alkanolamine-H2O datasets by robust neural network winsorization
Suleman, H., Maulud, A. S., & Man, Z. B. (2016) · Neural Computing and Applications · doi:10.1007/s00521-016-2213-z · Cited by 13
“Contrarily, robust estimation techniques like winsorization [42] retain the important information at the center and extremes of the system.”
BackgroundSmith, M. H., & Licklider, J. C. R. (1949) · Psychological Bulletin · doi:10.1037/h0056976 · Cited by 13
Church, J. A., Rukobo, S., Govha, M., et al. (2021) · EClinicalMedicine · doi:10.1016/j.eclinm.2021.101173 · Cited by 12
“truncation at the 5th and 95th centiles) [23].”
Dalai, C., & Dhar, A. (2022) · Flow Measurement and Instrumentation · doi:10.1016/j.flowmeasinst.2022.102298 · Cited by 12
Improving the Wisdom of Crowds with Analysis of Variance of Predictions of Related Outcomes
Satopää, V. (2021) · International Journal of Forecasting · doi:10.1016/j.ijforecast.2021.03.011 · Cited by 10
McFadden, D. T., Bauman, A., Hadrich, J. C., Jablonski, B. B. R., & Sullins, M. (2021) · Agricultural Finance Review · doi:10.1108/afr-05-2021-0070 · Cited by 12
Venkatesan, A., Forster, B., Rao, P., Miller, M., & Scahill, M. D. (2022) · JMIR Formative Research · doi:10.2196/38005 · Cited by 16
“In order to retain these heavy app users, but to limit their influence on downstream analyses, all usage factors were right-winsorized at the 99th percentile [54].”
BackgroundDoes Racial Discrimination Exist Within the NBA? An Analysis Based on Salary‐per‐Contribution*
Wen, R. (2018) · Social Science Quarterly · doi:10.1111/ssqu.12485 · Cited by 12
Quantifying the Managerial Ability of Microfinance Institutions: Evidence from Latin America
Banna, H., Rana, M. S., Ismail, I., et al. (2019) · Journal of International Development · doi:10.1002/jid.3419 · Cited by 12
The Wisdom of Multiple Guesses
Ugander, J. H., Drapeau, R., & Guestrin, C. (2015) · ACM Conference on Economics and Computation (EC) · doi:10.1145/2764468.2764529 · Cited by 11
“A possible estimation-based approach, not explored, would be to Winsorize the weights [Hastings et al. 1947].”
MethodOn the normality of transformed beta and unit- gamma random variables
Ratnaparkhl, M. V., & Mosimann, J. E. (1990) · Communication in Statistics- Theory and Methods · doi:10.1080/03610929008830416 · Cited by 11
gk: An R Package for the g-and-k and Generalised g-and-h Distributions
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“Work on quantile distributions goes back at least to Hastings et al. (1947).”
BackgroundMethodConfidence intervals for quantiles using generalized lambda distributions
Su, S. Y. S. (2009) · Computational Statistics & Data Analysis · doi:10.1016/j.csda.2009.02.014 · Cited by 11
“The Ramberg and Schmeiser (1974) (RS) GLD is an extension of Tukey’s lambda distribution (Hastings et al., 1947).”
Intangibilidade e o valor da empresa: uma análise do mercado acionário brasileiro
Gomes, H. B., Gonçalves, T. J. D. C., & Tavares, A. D. L. (2020) · Revista Catarinense da Ciência Contábil · doi:10.16930/2237-766220203045 · Cited by 11
Modeling and Fitting Quantile Distributions and Regressions
Gilchrist, W. G. (2007) · American Journal of Mathematical and Management Sciences · doi:10.1080/01966324.2007.10737707 · Cited by 11
“A paper by Hastings et al (1947) suggested a formula for”
MethodOn fitting cryptocurrency log-return exchange rates
Alzaatreh, A., & Sulieman, H. (2019) · Empirical Economics · doi:10.1007/s00181-019-01782-6 · Cited by 11
Optimally Robust Linear Estimators of Location
Birnbaum, A., LASKA, E. M., & Meisner, M. (1971) · Journal of the American Statistical Association · doi:10.1080/01621459.1971.10482259 · Cited by 11
Reproducible Analysis of Sequencing-Based RNA Structure Probing Data with User-Friendly Tools
Kielpinski, L. J., Sidiropoulos, N., & Vinther, J. (2015) · Methods in enzymology on CD-ROM/Methods in enzymology · doi:10.1016/bs.mie.2015.01.014 · Cited by 10
User-curated image collections: Modeling and recommendation
Li, Y., Mei, T., Cong, Y., et al. (2015) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata.2015.7363803 · Cited by 10
Validating Data Quality Actions in Scoring Processes
Cappiello, C., Cerletti, C., Fratto, C., & Pernici, B. (2017) · Journal of Data and Information Quality · doi:10.1145/3141248 · Cited by 10
“One possible method used to delete the outliers is Winsorization (Hastings et al. 1947), which consists of cutting values above the 95th percentile, to eliminate samples associated with faults characterized by a very high impact, and those below the 5th percentile, to eliminate samples associated with faults character…”
MethodThe Usefulness of the Two-Step Normality Transformation in Retesting Existing Theories
Templeton, G. F., Pope, M. B., & Burney, L. L. (2021) · ACM SIGMIS Database the DATABASE for Advances in Information Systems · doi:10.1145/3447934.3447939 · Cited by 11
“From the Original dataset, we generated three alternative data sets by applying the Two-Step, natural log (replicating the original study), and winsorization 5 (Hastings et al., 1947) transformations.”
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Ore, J., Detweiler, C., & Elbaum, S. G. (2021) · ACM Transactions on Software Engineering and Methodology · doi:10.1145/3439775 · Cited by 10
“To address the furthest outliers, we Winsorize [26] outliers to the 2nd/98th percentiles.”
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Şeref, O., Chaovalitwongse, W. A., & Brooks, J. P. (2012) · Annals of Operations Research · doi:10.1007/s10479-012-1193-3 · Cited by 10
“Winsorization is a procedure to reduce the effect of outliers by pushing all samples whose Mahalanobis distances are beyond 100 × ( 1 − p) th percentile to a distance equal to the 100 × ( 1 − p) th percentile towards the center (Hastings et al. 1947).”
MethodStatistical remarks on laboratory‐evaluating programs for comparing laboratories and methods
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Bain, L. J., & Thoman, D. R. (1968) · Journal of the American Statistical Association · doi:10.1080/01621459.1968.11009314 · Cited by 10
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John Wilder Tukey. 16 June 1915 – 26 July 2000
McCullagh, P. J. (2003) · Biographical Memoirs of Fellows of the Royal Society · doi:10.1098/rsbm.2003.0032 · Cited by 9
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Anscombe, F. J. (2007) · Springer eBooks · doi:10.1007/978-0-387-44956-2_42 · Cited by 9
On testing a shape parameter in the presence of a location and a scale parameter
van Montfort, M. A. J., & Otten, A. (1978) · Series Statistics · doi:10.1080/02331887808801411 · Cited by 9
Computationally Efficient Robust Estimation of Sparse FunctionalsPreprint
Du, S. S., Balakrishnan, S., & Singh, A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1702.07709 · Cited by 9
“…drawback of many of these classical robust estimators is that they are either heuristic in nature (for instance, methods based on Winsorization [Hastings Jr et al., 1947]) and are generally not optimal in the minimax sense, or are computationally intractable (for instance, methods based on…”
MethodSoil Microbiome Signatures are Associated with Pesticide Residues in Arable LandscapesPreprint
Walder, F., Schmid, M. W., Riedo, J., Valzano‐Held, A., Banerjee, S., Büchi, L., Bucheli, T. D., & van der Heijden, M. G. A. (2022) · SSRN Electronic Journal · doi:10.2139/ssrn.4146381 · Cited by 9
“In each model, factors were fitted sequentially (type I sum of squares), and the terms “ field-ID ” and “ fiel-d-ID:layer ” were used as random terms employing appropriate error terms and denominator degrees of freedom (Hastings et al., 1947).”
MethodMeasuring inequality and social welfare from any arbitrary distribution
Beena, V. T., & Kumaran, M. (2009) · Brazilian Journal of Probability and Statistics · doi:10.1214/08-bjps022 · Cited by 8
A new class of quantile functions useful in reliability analysis
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Ishigami, J., Honda, Y., Karger, A. B., et al. (2022) · Mayo Clinic Proceedings Innovations Quality & Outcomes · doi:10.1016/j.mayocpiqo.2022.03.002 · Cited by 8
Exchange Rates in Emerging Markets in the First Wave of the Covid-19 Pandemic
Kohrt, L., Horky, F. M., & Fidrmuc, J. (2022) · Eastern European Economics · doi:10.1080/00128775.2022.2074461 · Cited by 8
“…to not over or under-state the seasonal effects of certain months in which shocks occurred, we winsorized the data by replacing data points of each time series below respectively above the 2.5 th and the 97.5 th quantile with the values of the 2.5 th and 97.5 th quantile (Hastings et al. 1947).”
MethodInfluentialAn EMG-based approach toward the assessment of implicit self-esteem
Schliebener, M., Kraft, L., & Dufner, M. (2023) · Acta Psychologica · doi:10.1016/j.actpsy.2023.103868 · Cited by 9
Univariate and Bivariate Log-Topp-Leone Distribution Using Censored and Uncensored Datasets
Usman, A. U., Ishaq, A. I., Suleiman, A. A., et al. (2023) · doi:10.3390/iocma2023-14421 · Cited by 8
“Generating statistical distributions began with the use of systems of differential equations, as in [1], the method of generating systems of frequency curves and the quantile method introduced by [2].”
MethodPrivate Equity Real Estate: Asymmetric Effects of Gearing and Drivers of PerformancePreprint
Fuerst, F., Lim, W., & Matysiak, G. A. (2014) · SSRN Electronic Journal · doi:10.2139/ssrn.2519794 · Cited by 7
Testing for Benford's Law: A Monte Carlo Comparison of MethodsPreprint
Joenssen, D. W. (2014) · SSRN Electronic Journal · doi:10.2139/ssrn.2545243 · Cited by 8
“Tukey distribution Variates of the Tukey distribution [47] have the representing function Tα;λ(u) = αu λ − (1− u) (17) where u is a random variable uniformly distributed on the unit-interval.”
Prodel, E., Gondim, M. L., Rocha, H. N. M., Mira, P. A. D. C., & da Nóbrega, A. C. L. (2022) · Clinical Autonomic Research · doi:10.1007/s10286-022-00879-w · Cited by 7
“Outliers were identified via the interquartile range rule using a multiplier of 3, and therefore we used winsorization for the statistical analysis [24, 45].”
MethodA new family of quantile functions and its applications
Maladan, D. K., & Sankaran, P. G. (2020) · Communication in Statistics- Theory and Methods · doi:10.1080/03610926.2020.1713368 · Cited by 7
“A major development in portraying quantile functions to model statistical data is given by Hastings et al. (1947), who introduced a family of distributions by a quantile function.”
MethodPhotostress Recovery Time as a Potential Predictive Biomarker for Age-Related Macular Degeneration
Brandl, C., Zimmermann, M. E., Herold, J. M., Helbig, H., Stark, K. J., & Heid, I. M. (2023) · Translational Vision Science & Technology · doi:10.1167/tvst.12.2.15 · Cited by 7
The covariance matrix of normal order statistics
Davis, C. S., & Stephens, M. A. (1977) · Communications in Statistics - Simulation and Computation · doi:10.1080/03610917708812028 · Cited by 7
“The need for good tables of V, for many populations, was pointed out by Hastings et al. (1947) and the magnitude of the problem of exact calculation was also stressed; subsequently, series expansions for V.. have been given by Plackett (1958) and by David and Johnson (1954).”
BackgroundVisualization and quality control tools for large-scale multiplex tissue analysis in TissUUmaps3
Beháňová, A., Avenel, C., Andersson, A., Chelebian, E., Klemm, A. H., Leiss, L. W., Östman, A., & Wählby, C. (2023) · Biological Imaging · doi:10.1017/s2633903x23000053 · Cited by 6
Wealth and the health of older Black women in the United States
Sharma, A. (2018) · Health Promotion International · doi:10.1093/heapro/day053 · Cited by 6
Esposito, P., Kearns, M. M., Smith, K. B., Chandrasegaram, R., Kadamani, A. K., Gandelman, M., Liang, J., Nikpoor, N., Tompkins, T. A., & Ismail, N. (2022) · Comprehensive Psychoneuroendocrinology · doi:10.1016/j.cpnec.2022.100147 · Cited by 6
Esposito, P., Gandelman, M., Rodriguez, C., Liang, J., & Ismail, N. (2022) · Brain Behavior & Immunity - Health · doi:10.1016/j.bbih.2022.100543 · Cited by 7
The Robustness of the RESET Test to Non-Normal Error Terms
Mantalos, P., & Shukur, G. (2007) · Computational Economics · doi:10.1007/s10614-007-9100-8 · Cited by 6
“However, a more flexible and convenient distribution to generate samples and control not only the tail weights but also the skewness and kurtosis parameters is the generalized Tukey-Lambda, GLD ( Hastings et al. 1947; Ramberg et al. 1979; Ramberg and Schmeiser 1974) distribution.”
BackgroundRandomized Response Procedure for the Estimation of the Population Ratio using Ranked Set Sampling
Moreno, A. S., Bouza, C. N., Sautto, J. M., et al. (2016) · Journal of Mathematics and Statistics · doi:10.3844/jmssp.2016.107.114 · Cited by 6
Bello, G. (2014) · VCU Scholars Compass (Virginia Commonwealth University) · doi:10.25772/3hvw-zq89 · Cited by 7
“To construct the Health Status Metric (HSM), we use the Weighted Quantile Sum (WQS) methodology outlined in Carrico (2013) and Gennings et al.”
MethodSu, S. Y. S. (2011) · Journal of Modern Applied Statistical Methods · doi:10.22237/jmasm/1320120960 · Cited by 6
Hu, L., Lingler, J. H., Dabbs, A. D., Dew, M. A., & Sereika, S. M. (2017) · Clinical Transplantation · doi:10.1111/ctr.13030 · Cited by 6
The Small-Sample Variance of the Pitman Location Estimators
Hoaglin, D. C. (1975) · Journal of the American Statistical Association · doi:10.1080/01621459.1975.10480317 · Cited by 6
The influence of John Tukey's work in robust methods for chemometrics and environmetrics
Kafadar, K. (2002) · Chemometrics and Intelligent Laboratory Systems · doi:10.1016/s0169-7439(01)00190-3 · Cited by 6
JACKKNIFE TESTS FOR HETEROSCEDASTICITY IN THE GENERAL LINEAR MODEL
Giaccotto, C., & Sharma, S. C. (1988) · Australian Journal of Statistics · doi:10.1111/j.1467-842x.1988.tb00851.x · Cited by 6
Investigating experiential avoidance as a mechanism of action in a mindfulness intervention
Weinrib, A. (2011) · doi:10.17077/etd.ol97lozj · Cited by 6
Crafford, W. L. (2015) · SUNScholar (Stellenbosch University) · Cited by 6
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
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