{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T04:51:49Z","timestamp":1777438309939,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,30]],"date-time":"2021-01-30T00:00:00Z","timestamp":1611964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002724","name":"American University of Sharjah","doi-asserted-by":"publisher","award":["AUS Open Access Fund"],"award-info":[{"award-number":["AUS Open Access Fund"]}],"id":[{"id":"10.13039\/501100002724","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The traditional linear regression model that assumes normal residuals is applied extensively in engineering and science. However, the normality assumption of the model residuals is often ineffective. This drawback can be overcome by using a generalized normal regression model that assumes a non-normal response. In this paper, we propose regression models based on generalizations of the normal distribution. The proposed regression models can be used effectively in modeling data with a highly skewed response. Furthermore, we study in some details the structural properties of the proposed generalizations of the normal distribution. The maximum likelihood method is used for estimating the parameters of the proposed method. The performance of the maximum likelihood estimators in estimating the distributional parameters is assessed through a small simulation study. Applications to two real datasets are given to illustrate the flexibility and the usefulness of the proposed distributions and their regression models.<\/jats:p>","DOI":"10.3390\/e23020173","type":"journal-article","created":{"date-parts":[[2021,1,30]],"date-time":"2021-01-30T06:22:20Z","timestamp":1611987740000},"page":"173","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["On the Regression Model for Generalized Normal Distributions"],"prefix":"10.3390","volume":"23","author":[{"given":"Ayman","family":"Alzaatreh","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, American University of Sharjah, Sharjah P.O. Box 26666, United Arab Emirates"}]},{"given":"Mohammad","family":"Aljarrah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tafila Technical University, Tafila 66110, Jordan"}]},{"given":"Ayanna","family":"Almagambetova","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Amsterdam, 1098 XH Amsterdam, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9106-9916","authenticated-orcid":false,"given":"Nazgul","family":"Zakiyeva","sequence":"additional","affiliation":[{"name":"Zuse Institute Berlin, 14195 Berlin, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1098\/rsta.1895.0010","article-title":"Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material","volume":"186","author":"Pearson","year":"1895","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1093\/biomet\/36.1-2.149","article-title":"Systems of frequency curves generated by methods of translation","volume":"36","author":"Johnson","year":"1949","journal-title":"Biometrika"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1214\/aoms\/1177730388","article-title":"Low moments for small samples: A comparative study of order statistics","volume":"18","author":"Hastings","year":"1947","journal-title":"Ann. Stat."},{"key":"ref_5","unstructured":"Tukey, J.W. (1960). The Practical Relationship Between the Common Transformations of Percentages of Counts and Amounts, Statistical Techniques Research Group, Princeton University. Technical Report 36."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1002\/wics.1255","article-title":"Methods for generating families of univariate continuous distributions in the recent decades","volume":"5","author":"Lee","year":"2013","journal-title":"WIREs Comput. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1081\/STA-120003130","article-title":"The beta-normal distribution and its applications","volume":"31","author":"Eugene","year":"2002","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"85","DOI":"10.22237\/jmasm\/1083370200","article-title":"Beta-normal distribution: Bimodality properties and applications","volume":"3","author":"Famoye","year":"2004","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3722","DOI":"10.1080\/03610926.2011.568156","article-title":"On some properties of the beta normal distribution","volume":"41","author":"Rego","year":"2012","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1155\/S1024123X04403068","article-title":"The beta Gumbel distribution","volume":"4","author":"Nadarajah","year":"2004","journal-title":"Math. Probl. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1080\/03610920903366149","article-title":"Some results for beta Frechet distribution","volume":"40","author":"Cordeiro","year":"2011","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1590\/S0001-37652011000200002","article-title":"Closed form expressions for moments of the beta Weibull distribution","volume":"83","author":"Cordeiro","year":"2011","journal-title":"Ann. Braz. Acad. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"173","DOI":"10.22237\/jmasm\/1177992960","article-title":"Beta-Weibull distribution: Some properties and applications to censored data","volume":"6","author":"Lee","year":"2007","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2077","DOI":"10.1002\/sim.3598","article-title":"A new generalization of Weibull distribution with application to a breast cancer dataset","volume":"28","author":"Wahed","year":"2009","journal-title":"Stat. Med."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1080\/02331880801983876","article-title":"The beta-Pareto distribution","volume":"42","author":"Akinsete","year":"2008","journal-title":"Statistics"},{"key":"ref_16","unstructured":"De Morais, A.L. (2009). A Class of Generalized Beta Distributions, Pareto Power Series and Weibull Power Series Dissertation, Universidade Federal de Pernambuco."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.1016\/j.csda.2010.09.009","article-title":"The beta Burr XII distribution with application to lifetime data","volume":"55","author":"Paranaiba","year":"2011","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.stamet.2008.04.001","article-title":"Kumaraswamy\u2019s distribution: A beta type distribution with tractability advantages","volume":"6","author":"Jones","year":"2009","journal-title":"Stat. Methodol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1080\/00949650903530745","article-title":"A new family of generalized distributions","volume":"81","author":"Cordeiro","year":"2011","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1880","DOI":"10.1016\/j.csda.2011.11.015","article-title":"Generalized beta-generated distributions","volume":"56","author":"Alexander","year":"2012","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/j.stamet.2008.12.003","article-title":"On families of beta- and generalized gamma-generated distributions and associated inference","volume":"6","author":"Zografos","year":"2009","journal-title":"Stat. Methodol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s40300-013-0007-y","article-title":"A new method for generating families of continuous distributions","volume":"71","author":"Alzaatreh","year":"2013","journal-title":"Metron"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1186\/2195-5832-1-2","article-title":"On generating T-X family of distributions using quantile functions","volume":"1","author":"Aljarrah","year":"2014","journal-title":"J. Stat. Distrib. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1186\/2195-5832-1-16","article-title":"T-normal family of distribution: A new approach to generalize the normal distribution","volume":"1","author":"Alzaatreh","year":"2014","journal-title":"J. Stat. Distrib. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.csda.2013.07.035","article-title":"The gamma-normal distribution: Properties and applications","volume":"69","author":"Alzaatreh","year":"2014","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_26","first-page":"171","article-title":"A class of distributions which includes the normal ones","volume":"12","author":"Azzalini","year":"1985","journal-title":"Scand. J. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/s11749-006-0030-x","article-title":"Analyzing skewed data by power-normal model","volume":"17","author":"Gupta","year":"2008","journal-title":"Test"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1186\/s40488-019-0098-y","article-title":"A new extended normal regression model: Simulations and applications","volume":"6","author":"Lima","year":"2019","journal-title":"J. Stat. Distrib. Appl."},{"key":"ref_29","first-page":"1","article-title":"On fitting cryptocurrency log-return exchange rates","volume":"15","author":"Alzaatreh","year":"2019","journal-title":"Empir. Econ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"25","DOI":"10.2307\/2348376","article-title":"A quantile alternative for Kurtosis","volume":"37","author":"Moors","year":"1988","journal-title":"Statistician"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Galton, F. (1883). Enquiries into Human Faculty and Its Development, Macmillan.","DOI":"10.1037\/14178-000"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.asoc.2018.01.029","article-title":"Building real estate valuation models with comparative approach through case-based reasoning","volume":"65","author":"Yeh","year":"2018","journal-title":"Appl. Soft Comput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/173\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:17:31Z","timestamp":1760159851000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,30]]},"references-count":32,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["e23020173"],"URL":"https:\/\/doi.org\/10.3390\/e23020173","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,30]]}}}