{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:00:19Z","timestamp":1776844819213,"version":"3.51.2"},"reference-count":15,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T00:00:00Z","timestamp":1726617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Antofagasta"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The skew distribution has the characteristic of appropriately modeling asymmetric unimodal data. However, in practice, there are several cases in which the data present more than one mode. In the literature, it is possible to find a large number of authors who have studied extensions based on the skew distribution to model this type of data. In this article, a new family is introduced, consisting of a multimodal modification to the family of skew distributions. Using the methodology of the weighted version of a function, we perform the product of the density function of a family of skew distributions with a polynomial of degree 4, thus obtaining a more flexible model that allows modeling data sets, whose distribution contains at most three modes. The density function, some properties, moments, skewness coefficients, and kurtosis of this new family are presented. This study focuses on the particular cases of skew-normal and Laplace distributions, although it can be applied to any other distribution. A simulation study was carried out, to study the behavior of the model parameter estimates. Illustrations with real data, referring to medicine and environmental data, show the practical performance of the proposed model in the two particular cases presented.<\/jats:p>","DOI":"10.3390\/sym16091224","type":"journal-article","created":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T09:43:13Z","timestamp":1726652593000},"page":"1224","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A New Multimodal Modification of the Skew Family of Distributions: Properties and Applications to Medical and Environmental Data"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2921-3056","authenticated-orcid":false,"given":"Jimmy","family":"Reyes","sequence":"first","affiliation":[{"name":"Departamento de Estad\u00edstica y Ciencia de Datos, Facultad de Ciencias B\u00e1sicas, Universidad de Antofagasta, Antofagasta 1270300, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mario A.","family":"Rojas","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica y Ciencia de Datos, Facultad de Ciencias B\u00e1sicas, Universidad de Antofagasta, Antofagasta 1270300, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pedro L.","family":"Cort\u00e9s","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica y Ciencia de Datos, Facultad de Ciencias B\u00e1sicas, Universidad de Antofagasta, Antofagasta 1270300, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0534-1774","authenticated-orcid":false,"given":"Jaime","family":"Arru\u00e9","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica y Ciencia de Datos, Facultad de Ciencias B\u00e1sicas, Universidad de Antofagasta, Antofagasta 1270300, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,18]]},"reference":[{"key":"ref_1","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_2","first-page":"224","article-title":"Alpha-skew-normal distribution","volume":"29","year":"2010","journal-title":"Proyecciones"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1002\/env.1026","article-title":"Bimodal extension based on the skew-normal distribution with application to pollen data","volume":"22","author":"Salinas","year":"2011","journal-title":"Environmetrics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1080\/00949655.2017.1381698","article-title":"Bimodality based on the generalized skew-normal distribution","volume":"88","author":"Venegas","year":"2018","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1111\/j.1469-1809.1934.tb02105.x","article-title":"The effect of methods of ascertainment upon the estimation of frequencies","volume":"6","author":"Fisher","year":"1934","journal-title":"Ann. Eugen."},{"key":"ref_7","first-page":"311","article-title":"On discrete distributions arising out of methods of ascertainment","volume":"27","author":"Rao","year":"1965","journal-title":"Sankhy\u0101 Indian J. Stat. Ser. A"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"G\u00f3mez-D\u00e9niz, E., Arnold, B.C., Sarabia, J.M., and G\u00f3mez, H.W. (2021). Properties and Applications of a New Family of Skew Distributions. Mathematics, 9.","DOI":"10.3390\/math9010087"},{"key":"ref_9","unstructured":"G\u00f3mez-D\u00e9niz, E., Calder\u00edn-Ojeda, E., and Sarabia, J.M. (2023). Bimodal and Multimodal Extensions of the Normal and Skew Normal Distribution s. Stat. J., accepted and available on the internet."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Reyes, J., G\u00f3mez-D\u00e9niz, E., G\u00f3mez, H.W., and Calder\u00edn-Ojeda, E. (2021). A Bimodal Extension of the Exponential Distribution with Applications in Risk Theory. Symmetry, 13.","DOI":"10.3390\/sym13040679"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Reyes, J., Arru\u00e9, J., Leiva, V., and Martin-Barreiro, C. (2021). A New Birnbaum- Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine. Mathematics, 9.","DOI":"10.3390\/math9161891"},{"key":"ref_12","first-page":"271","article-title":"A probabilistic representation of the Skew-Normal distribution","volume":"4","author":"Henze","year":"1986","journal-title":"Scand. J. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/BF02294361","article-title":"The general theory and its analytical extension","volume":"52","author":"Bozdogan","year":"1974","journal-title":"Psychometrika"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1214\/aoms\/1177728074","article-title":"Tables for computing bivariate normal probabilities","volume":"27","author":"Owen","year":"1956","journal-title":"Ann. Math. Stat."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/9\/1224\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:58:28Z","timestamp":1760111908000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/9\/1224"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,18]]},"references-count":15,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["sym16091224"],"URL":"https:\/\/doi.org\/10.3390\/sym16091224","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,18]]}}}