{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,20]],"date-time":"2026-07-20T11:54:44Z","timestamp":1784548484744,"version":"3.55.0"},"reference-count":29,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T00:00:00Z","timestamp":1669248000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Universidad de Atacama","award":["ATA1956\u2013CC88433"],"award-info":[{"award-number":["ATA1956\u2013CC88433"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper presents an important theorem, which shows that, heading from the moments of the standard normal distribution, one can generate density functions originating a family of models. Additionally, we discussed that different random variable domains are achieved with transformations. For instance, we adopted the moment of order two, from the proposed theorem, and transformed it, which enabled us to exemplify this class as a unit distribution. We named it as Alpha-Unit (AU) distribution, which contains a single positive parameter \u03b1 (AU(\u03b1)\u2208[0,1]). We presented its properties and demonstrated two estimation methods for the \u03b1 parameter, the maximum likelihood estimator (MLE) and uniformly minimum-variance unbiased estimator (UMVUE) methods. In order to analyze the statistical consistency of the estimators, a Monte Carlo simulation study was carried out, in which the robustness was demonstrated. As a real-world application, we adopted two sets of unit data, the first regarding the dynamics of Chilean inflation in the post-military period, and the other one regarding the daily maximum relative humidity of the air in the Atacama Desert. In both cases presented, the AU model is competitive, whenever the data present a range greater than 0.4 and extremely heavy asymmetric tail. We compared our model with other commonly used unit models, such as the beta, Kumaraswamy, logit-normal, simplex, unit-half-normal, and unit-Lindley distributions.<\/jats:p>","DOI":"10.3390\/axioms11120666","type":"journal-article","created":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T03:00:13Z","timestamp":1669345213000},"page":"666","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Extending Normality: A Case of Unit Distribution Generated from the Moments of the Standard Normal Distribution"],"prefix":"10.3390","volume":"11","author":[{"given":"Miguel S.","family":"Concha-Aracena","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1530000, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leonardo","family":"Barrios-Blanco","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1530000, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Elal-Olivero","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1530000, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paulo Henrique","family":"Ferreira da Silva","sequence":"additional","affiliation":[{"name":"Department of Statistics, Federal University of Bahia, Salvador 40170110, Brazil"},{"name":"Centro de Pesquisa em Matem\u00e1tica Aplicada \u00e0 Ind\u00fastria (CeMEAI), University of S\u00e3o Paulo, S\u00e3o Carlos 13566590, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3406-4518","authenticated-orcid":false,"given":"Diego Carvalho do","family":"Nascimento","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1530000, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Fonseca, A., Ferreira, P.H., Nascimento, D.C.d., Fiaccone, R., Ulloa-Correa, C., Garc\u00eda-Pi\u00f1a, A., and Louzada, F. 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