{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T08:26:49Z","timestamp":1770539209820,"version":"3.49.0"},"reference-count":37,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,22]],"date-time":"2020-05-22T00:00:00Z","timestamp":1590105600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Marshall-Olkin extended family of distributions is an alternative for modeling lifetimes, and considers more or less asymmetry than its parent model, achieved by incorporating just one extra parameter. We investigate the bias of maximum likelihood estimators and use it to develop an estimator with less bias than traditional estimators, by a modification of the score function. Unlike other proposals, in this paper, we consider a bias reduction methodology that can be applied to any member of the family and not necessarily to any particular distribution. We conduct a Monte Carlo simulation in order to study the performance of the corrected estimators in finite samples. This simulation shows that the maximum likelihood estimator is quite biased and the proposed estimator is much less biased; in small sample sizes, the bias is reduced by around 50 percent. Two applications, related to the air conditioning system of an airplane and precipitations, are presented to illustrate the results. In those applications, the bias reduction for the shape parameters is close to 25% and the bias reduction also reduces, among others things, the width of the 95% confidence intervals for quantiles lower than 0.594.<\/jats:p>","DOI":"10.3390\/sym12050851","type":"journal-article","created":{"date-parts":[[2020,5,22]],"date-time":"2020-05-22T10:18:18Z","timestamp":1590142698000},"page":"851","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Bias Reduction for the Marshall-Olkin Extended Family of Distributions with Application to an Airplane\u2019s Air Conditioning System and Precipitation Data"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3814-9532","authenticated-orcid":false,"given":"Tiago M.","family":"Magalh\u00e3es","sequence":"first","affiliation":[{"name":"Department of Statistics, Institute of Exact Sciences, Federal University of Juiz de Fora, Juiz de Fora MG 36036-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yolanda M.","family":"G\u00f3mez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1530000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8184-7403","authenticated-orcid":false,"given":"Diego I.","family":"Gallardo","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1530000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6643-6972","authenticated-orcid":false,"given":"Osvaldo","family":"Venegas","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias Matem\u00e1ticasy F\u00edsicas, Facultad de Ingenier\u00eda, Universidad Cat\u00f3lica de Temuco, Temuco 4780000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1093\/biomet\/84.3.641","article-title":"A new method for adding a parameter to a family of distributions with application to the exponential and weibull families","volume":"84","author":"Marshall","year":"1997","journal-title":"Biometrika"},{"key":"ref_2","unstructured":"Marshall, A.W., and Olkin, I. (2007). Life Distributions. Structure of Nonparametric, Semipara- Metric and Parametric Families, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4537","DOI":"10.1080\/03610926.2014.922986","article-title":"On the Marshall-Olkin extended distributions","volume":"45","author":"Castellares","year":"2016","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_4","first-page":"17","article-title":"Marshall-Olkin extended pareto distribution and its application","volume":"18","author":"Ghitany","year":"2005","journal-title":"Int. J. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1080\/02664760500165008","article-title":"Marshall-Olkin extended Weibull distribution and its application to censored data","volume":"32","author":"Ghitany","year":"2005","journal-title":"J. Appl. Stat."},{"key":"ref_6","first-page":"107","article-title":"A marshall-olkin gamma distribution and minification process","volume":"1","author":"Ristic","year":"2007","journal-title":"STARS Int. J. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1855","DOI":"10.1080\/03610920601126571","article-title":"Marshall-olkin extended lomax distribution and its application to censored data","volume":"36","author":"Ghitany","year":"2007","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1017\/S0269964807000071","article-title":"Reliability properties of extended linear failure-rate distributions","volume":"21","author":"Ghitany","year":"2007","journal-title":"Probab. Eng. Inf. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s00362-006-0024-5","article-title":"On a generalization to Marshall\u2013Olkin scheme and its application to Burr type XII distribution","volume":"49","author":"Jayakumar","year":"2008","journal-title":"Stat. Pap."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2021","DOI":"10.1016\/j.csda.2010.03.003","article-title":"A new skew generalization of the normal distribution: Properties and applications","volume":"54","year":"2010","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/s11749-009-0169-3","article-title":"Another generalization of the geometric distribution","volume":"19","year":"2010","journal-title":"Test"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1214\/11-BJPS160","article-title":"A new extension of the Birnbaum-Saunders distribution","volume":"27","author":"Lemonte","year":"2013","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s00362-012-0431-8","article-title":"On the Marshall-Olkin extended Weibull distribution","volume":"54","author":"Cordeiro","year":"2013","journal-title":"Stat. Pap."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1080\/15598608.2013.802659","article-title":"The Marshall-Olkin family of distributions: Mathematical properties and new models","volume":"8","author":"Cordeiro","year":"2013","journal-title":"J. Stat. Theory Pract."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/j.cam.2014.04.020","article-title":"A new family of Marshall-Olkin extended distributions","volume":"271","author":"Alshangiti","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40488-015-0027-7","article-title":"The beta Marshall-Olkin family of distributions","volume":"2","author":"Alizadeh","year":"2015","journal-title":"J. Stat. Distrib. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s40300-014-0056-x","article-title":"Marshall-Olkin generalized exponential distribution","volume":"73","author":"Kundu","year":"2015","journal-title":"Metron"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"8306","DOI":"10.1080\/03610918.2016.1277747","article-title":"The Marshall-Olkin extended generalized Lindley distribution: Properties and applications","volume":"46","author":"Benkhelifa","year":"2017","journal-title":"Commun. Stat. Simul. Computation."},{"key":"ref_19","first-page":"365","article-title":"The Marshall-Olkin additive Weibull distribution with variable shapes for the hazard rate","volume":"47","author":"Afify","year":"2018","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"684","DOI":"10.1016\/j.jksus.2018.01.001","article-title":"The Marshall-Olkin Kappa distribution: Properties and applications","volume":"31","author":"Javed","year":"2019","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1080\/03610926.2017.1414254","article-title":"The Marshall-Olkin logistic-exponential distribution","volume":"48","author":"Mansoor","year":"2019","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_22","first-page":"273","article-title":"The Marshall-Olkin generalized-G family of distributions with applications","volume":"78","author":"Yousof","year":"2018","journal-title":"Statistica"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1649","DOI":"10.1029\/92WR00554","article-title":"Distributions of fractional order statistics in hydrology","volume":"28","author":"Durrans","year":"1992","journal-title":"Water Resour. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2238","DOI":"10.1080\/02664763.2016.1252730","article-title":"Marshall-Olkin distribution: Parameter estimation and application to cancer data","volume":"44","author":"Guney","year":"2017","journal-title":"J. Appl. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4171","DOI":"10.1080\/03610926.2018.1490430","article-title":"The Weibull Marshall-Olkin family: Regression model and application to censored data","volume":"48","author":"Korkmaz","year":"2019","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.cam.2018.10.052","article-title":"The Marshall-Olkin alpha power family of distributions with applications","volume":"1","author":"Nassar","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1111\/j.2517-6161.1968.tb00724.x","article-title":"A general definition of residuals","volume":"30","author":"Cox","year":"1968","journal-title":"J. R. Society. Ser. B Methodol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1093\/biomet\/80.1.27","article-title":"Bias reduction of maximum likelihood estimates","volume":"80","author":"Firth","year":"1993","journal-title":"Biometrika"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4259","DOI":"10.1016\/j.jspi.2005.08.043","article-title":"Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions","volume":"136","author":"Sartori","year":"2006","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Cordeiro, G.M., and Cribari-Neto, F. (2014). An introduction to Bartlett Corrections and Bias Reduction, Springer.","DOI":"10.1007\/978-3-642-55255-7"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2967","DOI":"10.1080\/00949655.2016.1143471","article-title":"Bias reduction of maximum likelihood estimates for a modified skew normal distribution","volume":"86","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_32","unstructured":"R Core Team (2017). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_33","unstructured":"Linhart, H., and Zucchini, W. (1986). Model Selection, John Wiley and Sons."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1080\/10618600.1996.10474708","article-title":"Randomized quantile residuals","volume":"5","author":"Dunn","year":"1996","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_35","first-page":"67","article-title":"On quick choice of power transformation","volume":"26","author":"Hinkley","year":"1977","journal-title":"Am. Stat."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4250","DOI":"10.1080\/03610926.2018.1490768","article-title":"Skewness of maximum likelihood estimators in the varying dispersion beta regression model","volume":"48","author":"Botter","year":"2019","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Magalh\u00e3es, T.M., Gallardo, D.I., and G\u00f3mez, H.W. (2019). Skewness of maximum likelihood estimators in the weibull censored data. Symmetry, 11.","DOI":"10.3390\/sym11111351"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/851\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:31:24Z","timestamp":1760175084000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/851"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,22]]},"references-count":37,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["sym12050851"],"URL":"https:\/\/doi.org\/10.3390\/sym12050851","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,22]]}}}