{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T01:28:38Z","timestamp":1776216518579,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T00:00:00Z","timestamp":1749772800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Graduate Studies and Scientific Research at Qassim University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper introduces a flexible distribution called the asymmetric double normal distribution, specifically designed to model univariate data exhibiting asymmetry and either unimodal or bimodal characteristics. This distribution is highly flexible, capable of capturing a wide range of data behaviors, from smooth densities to those with thinner tails. It generalizes the skew-normal distribution as a special case and provides a simpler alternative to mixture models by avoiding issues related to parameter identifiability. This study explores the structural and theoretical properties of the asymmetric double normal distribution, and parameter estimation is carried out using the maximum likelihood method. Simulation experiments assess the performance of the estimators, while applications in regression and real-life data fitting illustrate the practical relevance of this model. This proposed distribution proves to be a powerful tool for modeling asymmetric and bimodal data, offering significant advantages for statistical analysis in diverse applications.<\/jats:p>","DOI":"10.3390\/sym17060942","type":"journal-article","created":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T06:19:28Z","timestamp":1749795568000},"page":"942","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Modeling Bimodal and Skewed Data: Asymmetric Double Normal Distribution with Applications in Regression"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7302-7461","authenticated-orcid":false,"given":"Hugo S.","family":"Salinas","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1531772, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6441-5377","authenticated-orcid":false,"given":"Guillermo","family":"Mart\u00ednez-Fl\u00f3rez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas y Estad\u00edstica, Facultad de Ciencias B\u00e1sicas, Universidad de C\u00f3rdoba, Monter\u00eda 230002, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3189-0670","authenticated-orcid":false,"given":"Hassan S.","family":"Bakouch","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia"}]},{"given":"Lamia","family":"Alyami","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences and Arts, Najran University, P.O. Box 1988, Najran 11001, Saudi Arabia"}]},{"given":"Wilson E.","family":"Caimanque","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ingenier\u00eda, Universidad de Atacama, Copiap\u00f3 1531772, Chile"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1080\/00401706.1961.10489974","article-title":"The folded normal distribution","volume":"3","author":"Leone","year":"1961","journal-title":"Technometrics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"12","DOI":"10.3390\/math2010012","article-title":"On the folded normal distribution","volume":"2","author":"Tsagris","year":"2014","journal-title":"Mathematics"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Salinas, H., Bakouch, H., Qarmalah, N., and Mart\u00ednez-Fl\u00f3rez, G.A. (2023). Flexible class of two-piece normal distribution with a regression illustration to biaxial fatigue data. Mathematics, 11.","DOI":"10.3390\/math11051271"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1484","DOI":"10.1016\/j.jspi.2008.07.016","article-title":"Bayesian modeling using a class of bimodal skew-elliptical distributions","volume":"139","author":"Quintana","year":"2009","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"231","DOI":"10.6339\/JDS.2013.11(2).1142","article-title":"A folded normal slash distribution and its applications to non-negative measurements","volume":"11","author":"Gui","year":"2013","journal-title":"J. Data Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/B978-0-444-59428-0.00007-2","article-title":"Chapter 6\u2014Statistical Methods for Distributional Analysis","volume":"Volume 2","author":"Atkinson","year":"2015","journal-title":"Handbook of Income Distribution"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1080\/03610926.2014.882950","article-title":"Bimodal skew-symmetric normal distribution","volume":"45","author":"Hassan","year":"2016","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1080\/03610926.2013.765475","article-title":"Bimodal symmetric-asymmetric power-normal families","volume":"47","author":"Bolfarine","year":"2018","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_9","first-page":"279","article-title":"On multiple constraint skewed models","volume":"43","author":"Arnold","year":"2009","journal-title":"Stat. J. Theor. Appl. Stat."},{"key":"ref_10","first-page":"199","article-title":"Further results on a class of distributions which includes the normal ones","volume":"46","author":"Azzalini","year":"1986","journal-title":"Statistica"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1007\/s00170-003-2043-x","article-title":"Application of the generalized folded-normal distribution to the process capability measures","volume":"26","author":"Lin","year":"2005","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Alyami, L., Panda, D.K., and Das, S. (2023). Bayesian noise modelling for state estimation of the spread of COVID-19 in Saudi Arabia with extended Kalman filters. 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J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1080\/00949655.2015.1016434","article-title":"Random number generation and estimation with the bimodal asymmetric power-normal distribution","volume":"86","author":"Bolfarine","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1214\/aos\/1176346577","article-title":"The dip test of unimodality","volume":"13","author":"Hartigan","year":"1985","journal-title":"Ann. Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"320","DOI":"10.2307\/2347485","article-title":"Computation of the dip statistics to test for unimodality","volume":"34","author":"Hartigan","year":"1985","journal-title":"Appl. Stat."},{"key":"ref_20","unstructured":"Baddeley, B. (2025, April 06). Goftest: Classical Goodness-of-Fit Tests for Univariate Distributions. R Package Version 1.2-3. Available online: https:\/\/cran.r-project.org\/web\/packages\/goftest."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s10260-010-0135-y","article-title":"Influence diagnostics in the tobit censored response model","volume":"19","author":"Barros","year":"2010","journal-title":"Stat. Methods Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/S0167-9473(02)00104-4","article-title":"Influence diagnostics in generalized log-gamma regression models","volume":"42","author":"Ortega","year":"2003","journal-title":"Comput. Stat. 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