{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:15:13Z","timestamp":1760145313933,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T00:00:00Z","timestamp":1720656000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econometrics"],"abstract":"<jats:p>Non-negative distributions are important tools in various fields. Given the importance of achieving a good fit, the literature offers hundreds of different models, from the very simple to the highly flexible. In this paper, we consider the power\u2013Pareto model, which is defined by its quantile function. This distribution has three parameters, allowing the model to take different shapes, including symmetrical and left- and right-skewed. We provide different distributional characteristics and discuss parameter estimation. In addition to the already-known Maximum Likelihood and Least Squares of the logarithm of the order statistics estimation methods, we propose several additional methods. A simulation study and an application to two datasets are conducted to illustrate the performance of the estimation methods.<\/jats:p>","DOI":"10.3390\/econometrics12030020","type":"journal-article","created":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T11:30:55Z","timestamp":1720697455000},"page":"20","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Comparing Estimation Methods for the Power\u2013Pareto Distribution"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8628-7281","authenticated-orcid":false,"given":"Frederico","family":"Caeiro","sequence":"first","affiliation":[{"name":"Department of Mathematics and Center for Mathematics and Applications (NOVA Math), NOVA School of Science and Technology (NOVA FCT), 2829-516 Caparica, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0311-6888","authenticated-orcid":false,"given":"Mina","family":"Norouzirad","sequence":"additional","affiliation":[{"name":"Center for Mathematics and Applications (NOVA Math), NOVA School of Science and Technology (NOVA FCT), 2829-516 Caparica, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,11]]},"reference":[{"key":"ref_1","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\u2013Statistical Journal"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Beirlant, Jan, Goegebeur, Yuri, Segers, Johan, and Teugels, Jozef L. (2004). Statistics of Extremes: Theory and Applications, John Wiley & Sons.","DOI":"10.1002\/0470012382"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bhatti, Sajjad Haider, Hussain, Shahzad, Ahmad, Tanvir, Aslam, Muhammad, Aftab, Muhammad, and Raza, Muhammad Ali (2018). Efficient estimation of pareto model: Some modified percentile estimators. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0196456"},{"key":"ref_4","unstructured":"Bowley, Arthur L. (1901). Elements of Statistics, PS King & Son."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1214\/aoms\/1177731607","article-title":"Cumulative frequency functions","volume":"13","author":"Burr","year":"1942","journal-title":"The Annals of Mathematical Statistics"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Caeiro, Frederico, Martins, Ana P., and Sequeira, In\u00eas J. (2015). Finite sample behaviour of classical and quantile regression estimators for the pareto distribution. AIP Conference Proceedings, AIP Publishing LLC.","DOI":"10.1063\/1.4912753"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Caeiro, Frederico, and Mateus, Ayana (2023). A new class of generalized probability-weighted moment estimators for the pareto distribution. Mathematics, 11.","DOI":"10.3390\/math11051076"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"115347","DOI":"10.1016\/j.cam.2023.115347","article-title":"Reduced bias estimation of the shape parameter of the log-logistic distribution","volume":"436","author":"Caeiro","year":"2024","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"413","DOI":"10.3406\/ecoap.1977.4213","article-title":"A new model for personal income distribution: Specification and estimation","volume":"30","author":"Dagum","year":"1977","journal-title":"Economie Appliqu\u00e9"},{"key":"ref_10","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":"North American Actuarial Journal"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gilchrist, Warren (2000). Statistical Modelling with Quantile Functions, Chapman & Hall\/CRC.","DOI":"10.1201\/9781420035919"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/BF03263522","article-title":"Bonferroni and gini indices for various parametric families of distributions","volume":"68","author":"Giorgi","year":"2010","journal-title":"METRON"},{"key":"ref_13","unstructured":"Hall, David J. (1991). Repeat Variability in Instantaneously Released Heavy Gas Clouds\u2013Some Wind Tunnel Experiments, Warren Spring Laboratory. Technical report LR 804 (PA)."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1111\/j.1467-842X.2006.00426.x","article-title":"A new family of non-negative distributions","volume":"48","author":"Hankin","year":"2006","journal-title":"Australian & New Zealand Journal of Statistics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1080\/00401706.1959.10489870","article-title":"A graphical estimation of mixed weibull parameters in life-testing of electron tubes","volume":"1","author":"Kao","year":"1959","journal-title":"Technometrics"},{"key":"ref_16","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":"Quality & Quantity"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"8400130","DOI":"10.1155\/2022\/8400130","article-title":"Improved shape parameter estimation for the three-parameter log-logistic distribution","volume":"2022","author":"Mateus","year":"2022","journal-title":"Computational and Mathematical Methods"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Mehta, Navya Jayesh, and Yang, Fan (2022). Portfolio optimization for extreme risks with maximum diversification: An empirical analysis. Risks, 10.","DOI":"10.3390\/risks10050101"},{"key":"ref_19","first-page":"25","article-title":"A quantile alternative for kurtosis","volume":"37","author":"Moors","year":"1988","journal-title":"Journal of the Royal Statistical Society: Series D (The Statistician)"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2618","DOI":"10.1016\/j.jspi.2010.03.025","article-title":"L-moments of residual life","volume":"140","author":"Nair","year":"2010","journal-title":"Journal of Statistical Planning and Inference"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Nair, Narayanan  Unnikrishnan, Sankaran, Paduthol Godan, and Balakrishnan, Narayanaswamy (2013). Quantile-Based Reliability Analysis, Springer.","DOI":"10.1007\/978-0-8176-8361-0"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ndlovu, Thabani, and Chikobvu, Delson (2023). The generalised pareto distribution model approach to comparing extreme risk in the exchange rate risk of bitcoin\/us dollar and south african rand\/us dollar returns. Risks, 11.","DOI":"10.3390\/risks11060100"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1145\/355606.361888","article-title":"An approximate method for generating symmetric random variables","volume":"15","author":"Ramberg","year":"1972","journal-title":"Communications of the Association for Computing Machinery"},{"key":"ref_24","unstructured":"Reiss, Rolf-Dieter, and Thomas, Michael (2007). Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, Birkh\u00e4user Verlag."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"201","DOI":"10.2143\/AST.20.2.2005443","article-title":"Estimation in the pareto distribution","volume":"20","author":"Rytgaard","year":"1990","journal-title":"ASTIN Bulletin: The Journal of the IAA"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1080\/03610918.2013.867992","article-title":"A new quantile function with applications to reliability analysis","volume":"45","author":"Sankaran","year":"2016","journal-title":"Communications in Statistics-Simulation and Computation"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Schluter, Christian (2018). Top incomes, heavy tails, and rank-size regressions. Econometrics, 6.","DOI":"10.3390\/econometrics6010010"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Shakeel, Muhammad, Haq, Muhammad Ahsan ul, Hussain, Ijaz, Abdulhamid, Alaa Mohamd, and Faisal, Muhammad (2016). Comparison of two new robust parameter estimation methods for the power function distribution. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0162536"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1016\/j.spl.2012.02.005","article-title":"Quantile based entropy function","volume":"82","author":"Sunoj","year":"2012","journal-title":"Statistics & Probability Letters"},{"key":"ref_30","unstructured":"Tukey, John Wilder The Practical Relationship between the Common Transformations of Percentages and Counts and of Amounts, Statistical Techniques Research Group, Princeton University. Technical Report 36."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"71","DOI":"10.14257\/ijast.2013.59.06","article-title":"A note on modified estimators for the parameters of the power function distribution","volume":"59","author":"Zaka","year":"2013","journal-title":"International Journal of Advanced Science and Technology"}],"container-title":["Econometrics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2225-1146\/12\/3\/20\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:15:01Z","timestamp":1760109301000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2225-1146\/12\/3\/20"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,11]]},"references-count":31,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["econometrics12030020"],"URL":"https:\/\/doi.org\/10.3390\/econometrics12030020","relation":{},"ISSN":["2225-1146"],"issn-type":[{"type":"electronic","value":"2225-1146"}],"subject":[],"published":{"date-parts":[[2024,7,11]]}}}