{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T22:53:11Z","timestamp":1767912791808,"version":"3.49.0"},"reference-count":51,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,23]],"date-time":"2023-02-23T00:00:00Z","timestamp":1677110400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this paper, we investigate the performance of a variety of frequentist estimation techniques for the scale and shape parameters of the Lomax distribution. These methods include traditional methods such as the maximum likelihood estimator and the method of moments estimator. A version of the maximum likelihood estimator adjusted for bias is included as well. Furthermore, an alternative moment-based estimation technique, the L-moment estimator, is included, along with three different minimum distance estimators. The finite sample performances of each of these estimators are compared in an extensive Monte Carlo study. We find that no single estimator outperforms its competitors uniformly. We recommend one of the minimum distance estimators for use with smaller samples, while a bias-reduced version of maximum likelihood estimation is recommended for use with larger samples. In addition, the desirable asymptotic properties of traditional maximum likelihood estimators make them appealing for larger samples. We include a practical application demonstrating the use of the described techniques on observed data.<\/jats:p>","DOI":"10.3390\/computation11030044","type":"journal-article","created":{"date-parts":[[2023,2,24]],"date-time":"2023-02-24T02:32:12Z","timestamp":1677205932000},"page":"44","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Fitting the Lomax Distribution: A Comparison between Minimum Distance Estimators and Other Estimation Techniques"],"prefix":"10.3390","volume":"11","author":[{"given":"Thobeka","family":"Nombebe","sequence":"first","affiliation":[{"name":"School of Mathematical and Statistical Sciences, North-West University, Potchefstroom 2520, South Africa"}]},{"given":"James","family":"Allison","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, North-West University, Potchefstroom 2520, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9395-9808","authenticated-orcid":false,"given":"Leonard","family":"Santana","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, North-West University, Potchefstroom 2520, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4661-6180","authenticated-orcid":false,"given":"Jaco","family":"Visagie","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, North-West University, Potchefstroom 2520, South Africa"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1086\/250454","article-title":"The new theories of economics","volume":"4","author":"Pareto","year":"1897","journal-title":"J. Political Econ."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1007\/s42081-021-00115-1","article-title":"Likelihood analysis and stochastic EM algorithm for left truncated right censored data and associated model selection from the Lehmann family of life distributions","volume":"4","author":"Mitra","year":"2021","journal-title":"Jpn. J. Stat. Data Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"12","DOI":"10.22237\/jmasm\/1608553800","article-title":"JMASM 57: Bayesian Survival Analysis of Lomax Family Models with Stan (R)","volume":"19","author":"Abujarad","year":"2021","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_5","unstructured":"Atkinson, A.B., and Harrison, A.J. (1978). Distribution of Personal Wealth in Britain, Cambridge University Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1287\/opre.16.2.307","article-title":"The Pareto distribution as a queue service discipline","volume":"2","author":"Harris","year":"1968","journal-title":"Oper. Res."},{"key":"ref_7","first-page":"2153","article-title":"Optimum step-stress accelerated life testing for Lomax distribution","volume":"12","author":"Hassan","year":"2009","journal-title":"J. Appl. Sci. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1049\/ip-com:20045155","article-title":"Traffic characteristics of aggregated module downloads for mobile terminal reconfiguration","volume":"135","author":"Holland","year":"2006","journal-title":"IEE Proc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1080\/00401706.1987.10488243","article-title":"Parameter and quantile estimation for the generalized Pareto distribution","volume":"29","author":"Hosking","year":"1987","journal-title":"Technometrics"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1353","DOI":"10.1016\/j.jspi.2008.11.019","article-title":"Parameter estimation of the generalized Pareto distribution\u2014Part I","volume":"140","author":"Kotz","year":"2010","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ramos, P.L., Nascimento, D.C., Ferreira, P.H., Weber, K.T., Santos, T.E.G., and Louzada, F. (2019). Modelling traumatic brain injury lifetime data: Improved estimators for the generalised gamma distribution under small samples. PLoS ONE, 14.","DOI":"10.1371\/journal.pone.0221332"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1080\/02331888.2011.559657","article-title":"Comparison of estimation methods for the Weibull distribution","volume":"47","author":"Teimouri","year":"2013","journal-title":"Statistics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"768","DOI":"10.6339\/JDS.201910_17(4).0009","article-title":"Comparison of estimation methods for unit-gamma distribution","volume":"17","author":"Dey","year":"2019","journal-title":"J. Data Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1934","DOI":"10.1080\/03610926.2011.600506","article-title":"On the bias of the maximum likelihood estimator for the two-parameter Lomax distribution","volume":"42","author":"Giles","year":"2013","journal-title":"Commun. Stat. Methods"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1080\/23311835.2017.1279397","article-title":"Comparison of the robust parameters estimation methods for the two-parameters Lomax distribution","volume":"4","author":"Shakeel","year":"2017","journal-title":"Cogent Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s10687-005-6475-6","article-title":"Robust and efficient estimation for the generalized Pareto distribution","volume":"7","author":"Schucany","year":"2004","journal-title":"Extremes"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1093\/biomet\/85.3.549","article-title":"Robust and efficient estimation by minimising a density power divergence","volume":"85","author":"Basu","year":"1998","journal-title":"Biometrika"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Basu, A., Shioya, H., and Park, C. (2011). Statistical Inference: The Minimum Distance Approach, Chapman and Hall.","DOI":"10.1201\/b10956"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1080\/01621459.1981.10477701","article-title":"Minimum distance estimators for location and goodness of fit","volume":"76","author":"Boos","year":"1981","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"616","DOI":"10.1080\/01621459.1980.10477522","article-title":"Minimum distance and robust estimation","volume":"75","author":"Parr","year":"1980","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1080\/03610928108828100","article-title":"On minimum Cramer-von Mises-norm parameter estimation","volume":"10","author":"Parr","year":"1981","journal-title":"Commun. Stat. Methods"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"108677","DOI":"10.1016\/j.spl.2019.108677","article-title":"Objective Bayesian analysis for the Lomax distribution","volume":"159","author":"Ferreira","year":"2020","journal-title":"Stat. Probab. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"He, D., Sun, D., and Zhu, Q. (2022). Bayesian analysis for the Lomax model using noninformative priors. Stat. Theory Relat. Fields, 1\u20138.","DOI":"10.1080\/24754269.2022.2133466"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1080\/03610910600591925","article-title":"Bayesian estimation of the two-parameter gamma distribution","volume":"35","author":"Son","year":"2006","journal-title":"Commun. Stat. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1111\/j.2517-6161.1990.tb01775.x","article-title":"L-moments: Analysis and estimation of distributions using linear combinations of order statistic","volume":"52","author":"Hosking","year":"1990","journal-title":"J. R. Stat. Soc."},{"key":"ref_26","first-page":"301","article-title":"Maximum-likelihood estimation of the parameters of the generalized extreme-value distribution","volume":"34","author":"Hosking","year":"1985","journal-title":"J. R. Stat. Society. Ser. C Appl. Stat."},{"key":"ref_27","first-page":"105","article-title":"Probability weighted moments:Definition and relation to parameters of several distributions expressable in inverse form","volume":"52","author":"Greenwood","year":"1979","journal-title":"Water Resour. Res."},{"key":"ref_28","unstructured":"Asquith, W. (2011). Univariate Distributional Analysis with L-moment Statistics Using R, Create Space Independent Platform."},{"key":"ref_29","unstructured":"Deville, Y. (2016). Renext: Renewal Method for Extreme Values Extrapolation, Institut de Radioprotection et de S\u00fbret\u00e9 Nucl\u00e9aire. R package version 3.1-0."},{"key":"ref_30","unstructured":"R Core Team (2021). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_31","first-page":"79","article-title":"Maximum product of spacings estimation with application to the lognormal distribution","volume":"1","author":"Cheng","year":"1979","journal-title":"Math. Rep."},{"key":"ref_32","first-page":"92","article-title":"The maximum spacing method. An estimation method related to the maximum likelihood method","volume":"11","author":"Ranneby","year":"1984","journal-title":"Scand. J. Stat."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/BF02949797","article-title":"Estimation by the minimum distance method","volume":"5","author":"Wolfowitz","year":"1953","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1214\/aos\/1176343842","article-title":"Minimum Pareto distance estimates for parameter models","volume":"5","author":"Beran","year":"1977","journal-title":"Ann. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1214\/aos\/1176344125","article-title":"An efficient and robust adaptive estimator of location","volume":"6","author":"Beran","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1093\/imamat\/6.1.76","article-title":"The convergence of a class of double-rank minimization algorithms","volume":"6","author":"Broyden","year":"1970","journal-title":"J. Inst. Math. Its Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1093\/comjnl\/13.3.317","article-title":"A New Approach to Variable Metric Algorithms","volume":"13","author":"Fletcher","year":"1970","journal-title":"Comput. J."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1090\/S0025-5718-1970-0258249-6","article-title":"A Family of Variable Metric Updates Derived by Variational Means","volume":"24","author":"Goldfarb","year":"1970","journal-title":"Math. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1090\/S0025-5718-1970-0274029-X","article-title":"Conditioning of quasi-Newton methods for function minimization","volume":"24","author":"Shanno","year":"1970","journal-title":"Math. Comput."},{"key":"ref_40","unstructured":"Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis, Chapman & Hall, Inc."},{"key":"ref_41","unstructured":"Von Alven, W.H. (1964). Reliability Engineering, Prentice Hall."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Efron, B., and Tibshirani, R.J. (1993). An Introduction to the Bootstrap, Chapman & Hall, Inc.","DOI":"10.1007\/978-1-4899-4541-9"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v084.i05","article-title":"evmix: An R package for Extreme Value Mixture Modeling, Threshold Estimation and Boundary Corrected Kernel Density Estimation","volume":"84","author":"Hu","year":"2018","journal-title":"J. Stat. Softw."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10651-021-00525-0","article-title":"Modelling multivariate data using product copulas and minimum distance estimators: An exemplary application to ecological traits","volume":"29","author":"Liebscher","year":"2022","journal-title":"Environ. Ecol. Stat."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s00180-010-0203-7","article-title":"Copula parameter estimation by maximum-likelihood and minimum-distance estimator: A simulation study","volume":"26","author":"Weiss","year":"2011","journal-title":"Comput. Stat."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.2307\/2999632","article-title":"Autoregressive conditional duration: A new model for irregularly spaced transaction data","volume":"66","author":"Engle","year":"1998","journal-title":"Econometrica"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Thas, O. (2010). Comparing Distributions, Springer.","DOI":"10.1007\/978-0-387-92710-7"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s00362-017-0930-8","article-title":"Goodness-of-fit tests in conditional duration models","volume":"61","author":"Meintanis","year":"2020","journal-title":"Stat. Pap."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Bauwens, L., and Giot, P. (2001). Econometric Modelling of Stock Market Intraday Activity, Kluver Academic Publisher.","DOI":"10.1007\/978-1-4757-3381-5"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/j.ijforecast.2003.09.014","article-title":"A comparison of financial duration models via density forecasts","volume":"20","author":"Bauwens","year":"2005","journal-title":"Int. J. Forecast."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"De Luga, G., and Gallo, G.M. (2004). Mixture processes for financial intradaily durations. Stud. Nonlinear Dyn. Econom., 8.","DOI":"10.2202\/1558-3708.1223"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/3\/44\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:40:46Z","timestamp":1760121646000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/3\/44"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,23]]},"references-count":51,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["computation11030044"],"URL":"https:\/\/doi.org\/10.3390\/computation11030044","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,23]]}}}