{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T11:24:05Z","timestamp":1771673045755,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T00:00:00Z","timestamp":1716768000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Princess Nourah bint Abdulrahman University","award":["44-PRFA-P-142"],"award-info":[{"award-number":["44-PRFA-P-142"]}]},{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["44-PRFA-P-142"],"award-info":[{"award-number":["44-PRFA-P-142"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper introduces a new four-parameter additive model, named xgamma-Burr XII distribution, by considering two competing risks: the former has the xgamma distribution and the latter has the Burr XII distribution. A graphical description of the xgamma-Burr XII distribution is presented, including plots of the probability density function, hazard rate and reversed hazard rate functions. The xgamma-Burr XII density has different shapes such as decreasing, unimodal, approximately symmetric and decreasing-unimodal. The main statistical properties of the proposed model are studied. The unknown model parameters, reliability, hazard rate and reversed hazard rate functions are estimated via the maximum likelihood method. The asymptotic confidence intervals of the parameters, reliability function, hazard rate function and reversed hazard rate function are also obtained. A simulation study is carried out to evaluate the performance of the maximum likelihood estimates. In addition, three real data are applied to show the superiority of the xgamma-Burr XII distribution over some known distributions in real-life applications.<\/jats:p>","DOI":"10.3390\/sym16060659","type":"journal-article","created":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T10:14:07Z","timestamp":1716804847000},"page":"659","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The Additive Xgamma-Burr XII Distribution: Properties, Estimation and Applications"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7017-8336","authenticated-orcid":false,"given":"Hebatalla H.","family":"Mohammad","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0312-8731","authenticated-orcid":false,"given":"Faten S.","family":"Alamri","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":[{"role":"author","vocabulary":"crossref"}]},{"given":"Heba N.","family":"Salem","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Commerce, Al-Azhar University (Girls\u2019 Branch), Cairo 11765, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3247-2013","authenticated-orcid":false,"given":"Abeer A.","family":"EL-Helbawy","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Commerce, Al-Azhar University (Girls\u2019 Branch), Cairo 11765, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1002\/asmb.948","article-title":"Constructions and applications of lifetime distributions","volume":"29","author":"Lai","year":"2013","journal-title":"Appl. Stoch. Models Bus. Ind."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0951-8320(95)00149-2","article-title":"Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function","volume":"52","author":"Xie","year":"1995","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/S0951-8320(00)00066-1","article-title":"A new model with bathtub-shaped failure rate using an additive Burr XII distribution","volume":"70","author":"Wang","year":"2000","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.ress.2012.10.018","article-title":"A new modified Weibull distribution","volume":"111","author":"Almalki","year":"2013","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2592","DOI":"10.1080\/00949655.2013.797982","article-title":"The exponential-Weibull distribution","volume":"84","author":"Cordeiro","year":"2013","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.ress.2015.08.010","article-title":"An additive modified Weibull distribution","volume":"145","author":"He","year":"2016","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"43","DOI":"10.17713\/ajs.v45i3.107","article-title":"The log-logistic Weibull distribution with applications to lifetime data","volume":"45","author":"Oluyede","year":"2016","journal-title":"Austrian J. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1007\/s40304-016-0096-z","article-title":"An additive Perks-Weibull model with bathtub-shaped hazard rate function","volume":"4","author":"Singh","year":"2016","journal-title":"Commun. Math. Stat."},{"key":"ref_9","first-page":"118","article-title":"The Burr XII modified Weibull distribution: Model, Properties and Applications","volume":"10","author":"Mdlongwa","year":"2017","journal-title":"Electron. J. Appl. Stat. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s40304-019-00185-4","article-title":"A new extension of Chen distribution with applications to lifetime data","volume":"9","author":"Tarvirdizade","year":"2019","journal-title":"Commun. Math. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.ress.2019.03.014","article-title":"The log-normal modified Weibull distribution and its reliability implications","volume":"188","author":"Shakhatreh","year":"2019","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_12","first-page":"718","article-title":"Lomax-Weibull distribution with properties and applications in lifetime analysis","volume":"2020","author":"Osagie","year":"2020","journal-title":"Int. J. Math. Anal. Optim. Theory Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"447","DOI":"10.18187\/pjsor.v16i3.2957","article-title":"The flexible Weibull extension-Burr XII distribution: Model, properties and applications","volume":"16","author":"Kamal","year":"2020","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1002\/qre.2740","article-title":"An additive Chen-Weibull distribution and its applications in reliability modeling","volume":"37","author":"Thach","year":"2021","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1557","DOI":"10.1080\/03610926.2020.1732658","article-title":"A novel flexible additive Weibull distribution with real-life applications","volume":"50","author":"Khalil","year":"2021","journal-title":"Commun. Stat.\u2014Theory Methods"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"33","DOI":"10.5539\/ijsp.v10n4p33","article-title":"A new Lindley-Burr XII distribution: Model, Properties and Applications","volume":"10","author":"Makubate","year":"2021","journal-title":"Int. J. Stat. Probab."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ress.2022.108460","article-title":"A reliability and survival model for one and two failure modes system with applications to complete and censored datasets","volume":"223","author":"Abba","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3179","DOI":"10.1002\/qre.3119","article-title":"An additive power-transformed half-logistic model and its applications in reliability","volume":"38","author":"Xavier","year":"2022","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Thach, T.T. (2022). A three-component additive Weibull distribution and its reliability applications. Symmetry, 14.","DOI":"10.3390\/sym14071455"},{"key":"ref_20","first-page":"191","article-title":"The additive flexible Weibull extension-Lomax distribution: Properties and estimation with applications to COVID-19 data","volume":"28","author":"Salem","year":"2022","journal-title":"Acad. Period. Ref. J. AL-Azhar Univ."},{"key":"ref_21","first-page":"784","article-title":"The additive Perks distribution and its applications in reliability analysis","volume":"20","year":"2022","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"M\u00e9ndez-Gonz\u00e1lez, L.C., Rodr\u00edguez-Pic\u00f3n, L.A., P\u00e9rez-Olgu\u00edn, I.J.C., and Vidal Portilla, L.R. (2023). An additive Chen distribution with applications to lifetime data. Axioms, 12.","DOI":"10.3390\/axioms12020118"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"M\u00e9ndez-Gonz\u00e1lez, L.C., Rodr\u00edguez-Pic\u00f3n, L.A., Rodr\u00edguez Borb\u00f3n, M.I., and Sohn, H. (2023). The Chen\u2013Perks distribution: Properties and Reliability Applications. Mathematics, 11.","DOI":"10.3390\/math11133001"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"774","DOI":"10.22237\/jmasm\/1462077420","article-title":"The xgamma distribution: Statistical properties and application","volume":"15","author":"Sen","year":"2016","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"61","DOI":"10.6339\/JDS.201701_15(1).0004","article-title":"The quasi xgamma distribution with application in bladder cancer data","volume":"15","author":"Sen","year":"2017","journal-title":"Data Sci. J."},{"key":"ref_26","first-page":"43","article-title":"The weighted xgamma distribution: Properties and application","volume":"10","author":"Sen","year":"2017","journal-title":"J. Reliab. Stat. Stud."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"674","DOI":"10.2991\/jsta.2018.17.4.9","article-title":"On properties and applications of a two-parameter XGamma distribution","volume":"17","author":"Sen","year":"2018","journal-title":"J. Stat. Theory Appl."},{"key":"ref_28","first-page":"65","article-title":"The quasi xgamma- Poisson distribution: Properties and application","volume":"11","author":"Sen","year":"2018","journal-title":"J. Turk. Stat. Assoc."},{"key":"ref_29","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":"Ann. Math. Stat."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1948","DOI":"10.1080\/02664763.2020.1783520","article-title":"Exponentiated odd Chen-G family of distributions: Statistical properties, Bayesian and non-Bayesian estimation with applications","volume":"48","author":"Eliwa","year":"2021","journal-title":"J. Appl. Stat."},{"key":"ref_31","unstructured":"Wolfram Research, Inc. (2016). Mathematica, Version 11.0, Wolfram Research, Inc."},{"key":"ref_32","unstructured":"Meeker, W.Q., and Escobar, L.A. (1998). Statistical Methods for Reliability Data, Wiley."},{"key":"ref_33","unstructured":"Murthy, D.N.P., Xie, M., and Jiang, R. (1998). Weibull Models, John Wiley and Sons."},{"key":"ref_34","first-page":"444","article-title":"A new extension exponential distribution with applications of COVID-19 data","volume":"22","author":"Mubarak","year":"2021","journal-title":"J. Financ. Bus. Res."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/659\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:48:56Z","timestamp":1760107736000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/659"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,27]]},"references-count":34,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060659"],"URL":"https:\/\/doi.org\/10.3390\/sym16060659","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,27]]}}}