{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T04:08:57Z","timestamp":1777435737413,"version":"3.51.4"},"reference-count":36,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,16]],"date-time":"2023-04-16T00:00:00Z","timestamp":1681603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Vicerrector\u00eda de Investigaci\u00f3n y Postgrado de la Universidad de Atacama"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper introduces the inverse-power Muth power series model, which is a composition of the inverse-power Muth and the class of power series distributions. The use of the Bell distribution in this context is emphasized for the first time in the literature. Probability density, survival and hazard functions are studied, as well as their moments. Using the stochastic representation of the model, the maximum-likelihood estimators are implemented by the use of the expectation-maximization algorithm, while standard errors are calculated using Oakes\u2019 method. Monte Carlo simulation studies are conducted to show the performance of the maximum-likelihood estimators in finite samples. Two applications to real datasets are shown, where our proposal is compared with some models based on power series compositions.<\/jats:p>","DOI":"10.3390\/axioms12040383","type":"journal-article","created":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T02:02:59Z","timestamp":1681696979000},"page":"383","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Compound Class of Inverse-Power Muth and Power Series Distributions"],"prefix":"10.3390","volume":"12","author":[{"given":"Leonardo","family":"Barrios-Blanco","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, 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\u00e1tica, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1530000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H\u00e9ctor J.","family":"G\u00f3mez","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias Matem\u00e1ticas y F\u00edsicas, Facultad de Ingenier\u00eda, Universidad Cat\u00f3lica de Temuco, Temuco 4780000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1182-5193","authenticated-orcid":false,"given":"Marcelo","family":"Bourguignon","sequence":"additional","affiliation":[{"name":"Departamento de Estat\u00edstica, Universidade Federal do Rio Grande do Norte, Natal 59078-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1414","DOI":"10.1080\/03610918.2015.1005232","article-title":"The compound class of linear failure rate-power series distributions: Model, properties, and applications","volume":"46","author":"Mahmoudi","year":"2017","journal-title":"Commun.-Stat.-Simul. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1016\/j.csda.2012.09.009","article-title":"The compound class of extended Weibull power series distributions","volume":"58","author":"Silva","year":"2013","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3761","DOI":"10.1080\/03610926.2014.911904","article-title":"Gompertz-power series distributions","volume":"45","author":"Jafari","year":"2016","journal-title":"Commun.-Stat.-Theory Methods"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1214\/13-BJPS234","article-title":"The Burr XII power series distributions: A new compounding family","volume":"29","author":"Silva","year":"2015","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1080\/00949655.2015.1049949","article-title":"Inverse Weibull power series distributions: Properties and applications","volume":"86","author":"Shafiei","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"603","DOI":"10.18187\/pjsor.v13i3.2072","article-title":"The Exponential Pareto Power Series Distribution: Theory and Applications","volume":"13","author":"Elbatal","year":"2017","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"259","DOI":"10.6339\/JDS.201904_17(2).0002","article-title":"The compound class of Janardan-power series distributions: Properties and applications","volume":"17","author":"Shekari","year":"2019","journal-title":"J. Data Sci."},{"key":"ref_8","first-page":"47","article-title":"Compound power series distribution with negative multinomial summands: Characterisation and risk process","volume":"18","author":"Jordanova","year":"2020","journal-title":"Revstat"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1007\/s40745-018-0171-2","article-title":"The Generalized Burr XII Power Series Distributions with Properties and Applications","volume":"6","author":"Elbatal","year":"2019","journal-title":"Ann. Data Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Rivera, P.A., Calder\u00edn-Ojeda, E., Gallardo, D.I., and G\u00f3mez, H.W. (2021). A Compound Class of the Inverse Gamma and Power Series Distributions. Symmetry, 13.","DOI":"10.3390\/sym13081328"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3451","DOI":"10.1080\/02664763.2021.1951683","article-title":"Inverse Lindley power series distributions: A new compounding family and regression model with censored data","volume":"49","author":"Shakhatreh","year":"2022","journal-title":"J. Appl. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1998653","DOI":"10.1155\/2022\/1998653","article-title":"Inverse Exponentiated Lomax Power Series Distribution: Model, Estimation, and Application","volume":"2022","author":"Hassan","year":"2022","journal-title":"J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Aldahlan, M.A., Jamal, F., Chesneau, C., Elbatal, I., and Elgarhy, M. (2020). Exponentiated power generalized Weibull power series family of distributions: Properties, estimation and applications. PLoS ONE, 15.","DOI":"10.1371\/journal.pone.0230004"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"100004","DOI":"10.1016\/j.jcmds.2021.100004","article-title":"Statistical theory and practice of the inverse power Muth distribution","volume":"1","author":"Chesneau","year":"2021","journal-title":"J. Comput. Math. Data Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1214\/aoms\/1177729894","article-title":"A class of random variable with discrete distribution","volume":"21","author":"Noak","year":"1950","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"186","DOI":"10.3846\/13926292.2017.1289481","article-title":"The power muth distribution","volume":"22","year":"2017","journal-title":"Math. Model. Anal."},{"key":"ref_17","first-page":"401","article-title":"Reliability models with positive memory derived from the mean residual life function","volume":"2","author":"Muth","year":"1977","journal-title":"Theory Appl. Reliab."},{"key":"ref_18","unstructured":"Singh, S.V., Elgarhy, M., Ahmad, Z., Sharma, V.K., and Hamedani, G.G. (2021). Mathematical Modeling, Computational Intelligence Techniques, and Renewable Energy. Advances in Intelligent Systems and Computing, Springer."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.22436\/jnsa.011.10.06","article-title":"A new Muth generated family of distributions with applications","volume":"11","author":"Abdullah","year":"2018","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1211526","DOI":"10.1155\/2021\/1211526","article-title":"A new truncated Muth generated family of distributions with applications","volume":"2021","author":"Almarashi","year":"2021","journal-title":"Complexity"},{"key":"ref_21","unstructured":"Georg, M. (2020). R Package Version 0.6.6., R Foundation for Statistical Computing."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum likelihood from incomplete data via the EM algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J.R. Stat. Soc. Ser."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1111\/1467-9868.00188","article-title":"Direct calculation of the information matrix via the EM algorithm","volume":"61","author":"Oakes","year":"1999","journal-title":"J.R. Stat. Soc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3030","DOI":"10.1080\/03610926.2019.1679182","article-title":"Compound zero-truncated Poisson normal distribution and its applications","volume":"50","author":"Raqab","year":"2021","journal-title":"Commun.-Stat.-Theory Methods"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Technol. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6342","DOI":"10.1080\/03610918.2016.1202276","article-title":"A simplified estimation procedure based on the EM algorithm for the power series cure rate model","volume":"46","author":"Gallardo","year":"2017","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v090.i10","article-title":"bsamGP: An R Package for Bayesian Spectral Analysis Models Using Gaussian Process Priors","volume":"90","author":"Jo","year":"2019","journal-title":"J. Stat. Softw."},{"key":"ref_28","unstructured":"R Core Team (2022). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Available online: https:\/\/www.R-project.org\/."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"1","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_31","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_32","unstructured":"Murthy, D.P., Xie, M., and Jiang, R. (2003). Weibull Models, Wiley & Sons, Incorporated, John."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6978","DOI":"10.1080\/03610926.2020.1869782","article-title":"A useful variance decomposition for destructive Waring regression cure model with an application to HIV data","volume":"51","author":"Rodrigues","year":"2022","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Azimi, R., Esmailian, M., Gallardo, D.I., and G\u00f3mez, H.J. (2022). A New Cure Rate Model Based on Flory\u2013Schulz Distribution: Application to the Cancer Data. Mathematics, 10.","DOI":"10.3390\/math10244643"},{"key":"ref_35","first-page":"132","article-title":"A queuing model with state dependent services rates","volume":"12","author":"Conway","year":"1962","journal-title":"J. Ind. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Consul, P.C., and Famoye, F. (2006). Lagrangian Probability Distributions, Birkh\u00e4user.","DOI":"10.1002\/0471667196.ess1381.pub2"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/383\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:16:55Z","timestamp":1760123815000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/383"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,16]]},"references-count":36,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["axioms12040383"],"URL":"https:\/\/doi.org\/10.3390\/axioms12040383","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,16]]}}}