{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:53:53Z","timestamp":1760147633994,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T00:00:00Z","timestamp":1676937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Funds through FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UIDB\/00297\/2020","UIDP\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020","UIDP\/00297\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Estimation based on probability-weighted moments is a well-established method and an excellent alternative to the classic method of moments or the maximum likelihood method, especially for small sample sizes. In this research, we developed a new class of estimators for the parameters of the Pareto type I distribution. A generalization of the probability-weighted moments approach is the foundation for this new class of estimators. It has the advantage of being valid in the entire parameter space of the Pareto distribution. We established the asymptotic normality of the new estimators and applied them to simulated and real datasets in order to illustrate their finite sample behavior. The results of comparisons with the most used estimation methods were also analyzed.<\/jats:p>","DOI":"10.3390\/math11051076","type":"journal-article","created":{"date-parts":[[2023,2,22]],"date-time":"2023-02-22T01:39:47Z","timestamp":1677029987000},"page":"1076","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A New Class of Generalized Probability-Weighted Moment Estimators for the Pareto Distribution"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8628-7281","authenticated-orcid":false,"given":"Frederico","family":"Caeiro","sequence":"first","affiliation":[{"name":"NOVA School of Science and Technology (FCT NOVA) and CMA, Campus de Caparica, NOVA University Lisbon, 2829-516 Caparica, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5630-3321","authenticated-orcid":false,"given":"Ayana","family":"Mateus","sequence":"additional","affiliation":[{"name":"NOVA School of Science and Technology (FCT NOVA) and CMA, Campus de Caparica, NOVA University Lisbon, 2829-516 Caparica, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,21]]},"reference":[{"key":"ref_1","unstructured":"Pareto, V. (1897). Cours d\u2019Economie Politique, Librairie Droz."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kleiber, C., and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, John Wiley & Sons.","DOI":"10.1002\/0471457175"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/10920277.2006.10596236","article-title":"Pareto Tail Index Estimation Revisited","volume":"10","author":"Finkelstein","year":"2006","journal-title":"N. Am. Actuar. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1080\/01621459.1954.10501239","article-title":"Business Failures: Another Example of the Analysis of Failure Data","volume":"49","author":"Lomax","year":"1954","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bourguignon, M., Gallardo, D.I., and G\u00f3mez, H.J. (2022). A Note on Pareto-Type Distributions Parameterized by Its Mean and Precision Parameters. Mathematics, 10.","DOI":"10.3390\/math10030528"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10888-021-09514-6","article-title":"Pareto models for top incomes and wealth","volume":"20","author":"Charpentier","year":"2022","journal-title":"J. Econ. Inequal."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Beirlant, J., Goegebeur, Y., Segers, J., and Teugels, J.L. (2004). Statistics of Extremes: Theory and Applications, John Wiley & Sons.","DOI":"10.1002\/0470012382"},{"key":"ref_8","first-page":"1","article-title":"An overview and open research topics in statistics of univariate extremes","volume":"10","author":"Beirlant","year":"2012","journal-title":"Revstat-Stat. J."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Albrecher, H., Beirlant, J., and Teugels, J.L. (2017). Reinsurance: Actuarial and Statistical Aspects, John Wiley & Sons, Ltd.","DOI":"10.1002\/9781119412540"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1111\/insr.12058","article-title":"Extreme value theory and statistics of univariate extremes: A review","volume":"83","author":"Gomes","year":"2015","journal-title":"Int. Stat. Rev."},{"key":"ref_11","unstructured":"Peng, L., and Qi, Y. (2017). Inference for Heavy-Tailed Data: Applications in Insurance and Finance, Academic Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF02613419","article-title":"Old and new methods of estimation and the Pareto distribution","volume":"10","author":"Quandt","year":"1966","journal-title":"Metrika"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1007\/s11135-007-9100-8","article-title":"The Estimation of Pareto Distribution by a Weighted Least Square Method","volume":"41","author":"Lu","year":"2007","journal-title":"Qual. Quant."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"540007","DOI":"10.1063\/1.4912753","article-title":"Finite sample behaviour of classical and quantile regression estimators for the Pareto distribution","volume":"1648","author":"Caeiro","year":"2015","journal-title":"AIP Conf. Proc."},{"key":"ref_15","first-page":"263","article-title":"Generalized least squares and weighted least squares estimation methods for distributional parameters","volume":"13","author":"Kantar","year":"2015","journal-title":"REVSTAT-Stat. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1016\/j.jkss.2017.01.004","article-title":"Parameter estimation of the Pareto distribution using a pivotal quantity","volume":"46","author":"Kim","year":"2017","journal-title":"J. Korean Stat. Soc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1080\/10920277.2000.10595935","article-title":"Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution","volume":"4","author":"Brazauskas","year":"2000","journal-title":"N. Am. Actuar. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"6252","DOI":"10.1016\/j.csda.2007.01.003","article-title":"A robust estimator for the tail index of Pareto-type distributions","volume":"51","author":"Vandewalle","year":"2007","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1080\/01621459.1989.10478875","article-title":"Bayesian estimation and prediction for Pareto data","volume":"84","author":"Arnold","year":"1989","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_20","first-page":"20","article-title":"Bayes estimators for the shape parameter of Pareto type I distribution under generalized square error loss function","volume":"4","author":"Rasheed","year":"2014","journal-title":"Math. Theory Model."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1834","DOI":"10.1080\/00949655.2020.1750612","article-title":"The E-Bayesian estimation and its E-MSE of Pareto distribution parameter under different loss functions","volume":"90","author":"Han","year":"2020","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF00872461","article-title":"Parameter estimations for 2-parameter Pareto distribution by pome","volume":"9","author":"Singh","year":"1995","journal-title":"Water Resour. Manag."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1016\/j.jspi.2010.08.015","article-title":"Semi-parametric tail inference through probability-weighted moments","volume":"141","author":"Caeiro","year":"2011","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_24","unstructured":"Caeiro, F., and Gomes, M.I. (2013). Recent Developments in Modeling and Applications in Statistics, Springer."},{"key":"ref_25","first-page":"45","article-title":"Comparison of different methods of parameters estimation for Pareto Model","volume":"2","author":"Munir","year":"2013","journal-title":"Casp. J. Appl. Sci. Res."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Bhatti, S.H., Hussain, S., Ahmad, T., Aslam, M., Aftab, M., and Raza, M.A. (2018). Efficient estimation of Pareto model: Some modified percentile estimators. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0196456"},{"key":"ref_27","first-page":"605","article-title":"Efficient estimation of Pareto model using modified maximum likelihood estimators","volume":"26","author":"Bhatti","year":"2019","journal-title":"Sci. Iran."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1195","DOI":"10.1007\/s00362-019-01132-9","article-title":"Pareto parameters estimation using moving extremes ranked set sampling","volume":"62","author":"Chen","year":"2019","journal-title":"Stat. Pap."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1029\/WR015i005p01049","article-title":"Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form","volume":"15","author":"Greenwood","year":"1979","journal-title":"Water Resour. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1080\/00401706.1985.10488049","article-title":"Estimation of the generalized extreme-value distribution by the method of probability-weighted moments","volume":"27","author":"Hosking","year":"1985","journal-title":"Technometrics"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/0022-1694(89)90191-1","article-title":"Expressions relating probability weighted moments to parameters of several distributions inexpressible in inverse form","volume":"110","author":"Jing","year":"1989","journal-title":"J. Hydrol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1029\/WR015i005p01055","article-title":"Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles","volume":"15","author":"Landwehr","year":"1979","journal-title":"Water Resour. Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1361","DOI":"10.1029\/WR015i006p01361","article-title":"Estimation of parameters and quantiles of Wakeby distributions: 1. Known lower bounds","volume":"15","author":"Landwehr","year":"1979","journal-title":"Water Resour. Res."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Bispo, R., Henriques-Rodrigues, L., Alpizar-Jara, R., and de Carvalho, M. (2021, January 13\u201316). Computational Study of the Adaptive Estimation of the Extreme Value Index with Probability Weighted Moments. Proceedings of the Recent Developments in Statistics and Data Science: SPE2021, \u00c9vora, Portugal.","DOI":"10.1007\/978-3-031-12766-3"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11009-012-9295-6","article-title":"Semi-parametric probability-weighted moments estimation revisited","volume":"16","author":"Caeiro","year":"2014","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1029\/2001WR900014","article-title":"Generalized probability weighted moments: Application to the generalized Pareto distribution","volume":"37","author":"Rasmussen","year":"2001","journal-title":"Water Resour. Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/978-3-319-18029-8_22","article-title":"A Log Probability Weighted Moment Estimator of Extreme Quantiles","volume":"Volume 136","author":"Kitsos","year":"2015","journal-title":"Theory and Practice of Risk Assessment"},{"key":"ref_38","unstructured":"Skiadas, C.H. (2017, January 6\u20139). Log Probability Weighted Moments Method for Pareto distribution. Proceedings of the 17th Applied Stochastic Models and Data Analysis International Conference with the 6th Demographics Workshop, London, UK."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"7761","DOI":"10.1080\/03610918.2016.1249884","article-title":"Parameter estimation for generalized Pareto distribution by generalized probability weighted moment-equations","volume":"46","author":"Chen","year":"2017","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"e1133","DOI":"10.1002\/cmm4.1133","article-title":"A new class of estimators for the shape parameter of a Pareto model","volume":"3","author":"Mateus","year":"2021","journal-title":"Comput. Math. Methods"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"320003","DOI":"10.1063\/5.0081541","article-title":"Confidence intervals for the shape parameter of a Pareto distribution","volume":"2425","author":"Mateus","year":"2022","journal-title":"AIP Conf. Proc."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Chotikapanich, D. (2008). Modeling Income Distributions and Lorenz Curves, Springer.","DOI":"10.1007\/978-0-387-72796-7"},{"key":"ref_43","unstructured":"Arnold, B.C., Balakrishnan, N., and Nagaraja, H.N. (1992). A First Course in Order Statistics, Siam."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1323","DOI":"10.1029\/WR024i008p01323","article-title":"Assessment of flood frequency models using empirical distribution function statistics","volume":"24","author":"Ahmad","year":"1988","journal-title":"Water Resour. Res."},{"key":"ref_45","first-page":"21","article-title":"Power Comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling Tests","volume":"2","author":"Razali","year":"2011","journal-title":"J. Stat. Model. Anal."},{"key":"ref_46","first-page":"29","article-title":"Goodness of Fit Tests and Power Comparisons for Weighted Gamma Distribution","volume":"14","author":"Singla","year":"2016","journal-title":"REVSTAT-Stat. J."},{"key":"ref_47","unstructured":"(2021, May 15). The 150 Largest Cities in the World. Available online: https:\/\/www.worldatlas.com\/citypops.htm."},{"key":"ref_48","unstructured":"Bowley, A.L. (1901). Elements of Statistics, PS King & Son."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1093\/biomet\/77.3.631","article-title":"Robust quantile estimators for skewed populations","volume":"77","author":"Horn","year":"1990","journal-title":"Biometrika"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/S1544-6123(03)00003-5","article-title":"On more robust estimation of skewness and kurtosis","volume":"1","author":"Kim","year":"2004","journal-title":"Financ. Res. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1198\/106186004X12632","article-title":"A Robust Measure of Skewness","volume":"13","author":"Brys","year":"2004","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"5947","DOI":"10.1016\/j.physa.2013.07.061","article-title":"Are your data really Pareto distributed?","volume":"392","author":"Cirillo","year":"2013","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_53","first-page":"1","article-title":"A Note on the Upper-truncated Pareto distribution","volume":"Winter 1","author":"Clark","year":"2013","journal-title":"Casualty Actuar. Soc. E-Forum"}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/5\/1076\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:38:11Z","timestamp":1760121491000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/5\/1076"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,21]]},"references-count":53,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["math11051076"],"URL":"https:\/\/doi.org\/10.3390\/math11051076","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2023,2,21]]}}}