{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T07:55:35Z","timestamp":1781250935909,"version":"3.54.1"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T00:00:00Z","timestamp":1694995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["PNURSP2023R226"],"award-info":[{"award-number":["PNURSP2023R226"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In many research fields, statistical probability models are often used to analyze real-world data. However, data from many fields, such as the environment, economics, and health care, do not necessarily fit traditional models. New empirical models need to be developed to improve the fit. In this study, we investigated a further extension of the quasi-Lindley model. This extension was asymmetrically distributed on the positive real number line. Maximum likelihood, least square error, Anderson\u2013Darling, and expectation maximization algorithms were used to estimate the parameters studied. All techniques provided accurate and reliable estimates of the parameters. However, the mean square error of the expectation-maximization approach was lower. The usefulness of the proposed model was demonstrated by analyzing a reliability data set, and the analysis showed that it outperformed all other alternative models.<\/jats:p>","DOI":"10.3390\/sym15091780","type":"journal-article","created":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T05:59:06Z","timestamp":1695016746000},"page":"1780","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Scaled-Invariant Extended Quasi-Lindley Model: Properties, Estimation, and Application"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6363-1246","authenticated-orcid":false,"given":"Mohamed","family":"Kayid","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdulrahman","family":"Abouammoh","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7856-6579","authenticated-orcid":false,"given":"Ghadah","family":"Alomani","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1111\/j.2517-6161.1958.tb00278.x","article-title":"Fiducial distributions and Bayes\u2019 theorem","volume":"20","author":"Lindley","year":"1958","journal-title":"J. R. Stat. Soc. Ser."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1016\/j.matcom.2007.06.007","article-title":"Lindley distribution and its application","volume":"78","author":"Ghitany","year":"2008","journal-title":"Math. Comput. Simul."},{"key":"ref_3","first-page":"143","article-title":"A New Quasi Lindley Distribution","volume":"8","author":"Shanker","year":"2013","journal-title":"Int. J. Stat. Syst."},{"key":"ref_4","first-page":"13","article-title":"Generalized Lindley distribution","volume":"3","author":"Zakerzadeh","year":"2009","journal-title":"J. Math. Ext."},{"key":"ref_5","first-page":"15","article-title":"A Three-Parameter Lindley Distribution","volume":"7","author":"Shanker","year":"2017","journal-title":"Am. J. Math. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"198951","DOI":"10.1155\/2014\/198951","article-title":"The Beta-Lindley Distribution: Properties and Applications","volume":"2014","author":"Merovci","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_7","first-page":"30","article-title":"A new generalized Lindley distribution","volume":"3","author":"Ibrahim","year":"2013","journal-title":"Math. Theory Model."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"145","DOI":"10.2307\/2529053","article-title":"The discrete poisson-Lindley distribution","volume":"26","author":"Sankaran","year":"1970","journal-title":"Biometrics"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.csda.2013.02.026","article-title":"Power Lindley distribution and associated inference","volume":"64","author":"Ghitany","year":"2013","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1080\/03610926.2011.563011","article-title":"Inferences on stress-strength reliability from Lindley distribution","volume":"42","author":"Ghitany","year":"2013","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4","DOI":"10.3844\/jmssp.2010.4.9","article-title":"Negative Binomial-Lindley Distribution and Its Application","volume":"6","author":"Zamani","year":"2010","journal-title":"J. Math. Stat."},{"key":"ref_12","first-page":"363","article-title":"The five parameter Lindley distribution","volume":"31","author":"Eid","year":"2014","journal-title":"Pak. J. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1080\/03610918.2013.767910","article-title":"Estimation of the reliability of a stress-strength system from power Lindley distributions","volume":"44","author":"Ghitany","year":"2015","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3662","DOI":"10.1080\/00949655.2014.995101","article-title":"A new generalized Lindley distribution","volume":"85","author":"Abouammoh","year":"2015","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_15","first-page":"473","article-title":"A new extension of Lindley distribution: Modified validation test, characterizations and different methods of estimation","volume":"26","author":"Ibrahim","year":"2019","journal-title":"Commun. Stat. Appl. Methods"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2049501","DOI":"10.1155\/2020\/2049501","article-title":"Generalized Weibull-Lindley (GWL) distribution in modeling lifetime Data","volume":"2020","author":"Marthin","year":"2020","journal-title":"J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Al-Babtain, A.A., Ahmed, A.H.N., and Afify, A.Z. (2020). A new discrete analog of the continuous Lindley distribution, with reliability applications. Entropy, 22.","DOI":"10.3390\/e22060603"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"28","DOI":"10.22271\/maths.2020.v5.i6a.610","article-title":"Lindley Gompertz distribution with properties and applications","volume":"5","author":"Joshi","year":"2020","journal-title":"Int. J. Appl. Math. Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1080\/16583655.2020.1715017","article-title":"The Weibull Marshall and Olkin Lindley distribution: Properties and estimation","volume":"14","author":"Afify","year":"2020","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/s42519-020-00116-5","article-title":"The Inverted Modified Lindley Distribution","volume":"14","author":"Chesneau","year":"2020","journal-title":"J. Stat. Theory Pract."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Algarni, A. (2021). On a new generalized lindley distribution: Properties, estimation and applications. PLoS ONE, 16.","DOI":"10.1371\/journal.pone.0244328"},{"key":"ref_22","first-page":"64","article-title":"A quasi Lindley distribution","volume":"6","author":"Shanker","year":"2013","journal-title":"Afr. J. Math. Comput. Sci. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"8467291","DOI":"10.1155\/2022\/8467291","article-title":"EM Algorithm for Estimating the Parameters of Quasi-Lindley Model with Application","volume":"2022","author":"Kayid","year":"2022","journal-title":"J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3747753","DOI":"10.1155\/2021\/3747753","article-title":"A New Scale-Invariant Lindley Extension Distribution and Its Applications","volume":"2021","author":"Kayid","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"101714","DOI":"10.1016\/j.jksus.2021.101714","article-title":"A new flexible extension of the Lindley distribution with applications","volume":"34","author":"Alrasheedi","year":"2022","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_26","unstructured":"Titterington, D.M., Smith, A.F.M., and Makov, U.E. (1985). Statistical Analysis of Finite Mixture Distributions, John Wiley and Sons."},{"key":"ref_27","unstructured":"Ord, J.K. (1972). Families of Frequency Distributions, Charles Griffin."},{"key":"ref_28","unstructured":"Lai, C.D., and Xie, M. (2006). Stochastic Aging and Dependence for Reliability, Springer."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1080\/00401706.1963.10490105","article-title":"Theoretical Explanation of Observed Decreasing Failure Rate","volume":"5","author":"Proschan","year":"1963","journal-title":"Technometrics"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1214\/aos\/1176346060","article-title":"On the convergence properties of the EM algorithm","volume":"11","author":"Wu","year":"1983","journal-title":"Ann. Stat."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1780\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:52:50Z","timestamp":1760129570000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1780"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,18]]},"references-count":30,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091780"],"URL":"https:\/\/doi.org\/10.3390\/sym15091780","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,18]]}}}