{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:12:38Z","timestamp":1760238758063,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,5]],"date-time":"2020-09-05T00:00:00Z","timestamp":1599264000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A new family of probability distributions is defined and applied for modeling symmetric real-life datasets. Some new bivariate type G families using Farlie\u2013Gumbel\u2013Morgenstern copula, modified Farlie\u2013Gumbel\u2013Morgenstern copula, Clayton copula and Renyi\u2019s entropy copula are derived. Moreover, some of its statistical properties are presented and studied. Next, the maximum likelihood estimation method is used. A graphical assessment based on biases and mean squared errors is introduced. Based on this assessment, the maximum likelihood method performs well and can be used for estimating the model parameters. Finally, two symmetric real-life applications to illustrate the importance and flexibility of the new family are proposed. The symmetricity of the real data is proved nonparametrically using the kernel density estimation method.<\/jats:p>","DOI":"10.3390\/sym12091462","type":"journal-article","created":{"date-parts":[[2020,9,6]],"date-time":"2020-09-06T23:12:49Z","timestamp":1599433969000},"page":"1462","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A New Parametric Life Family of Distributions: Properties, Copula and Modeling Failure and Service Times"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3456-8393","authenticated-orcid":false,"given":"Mansour","family":"Shrahili","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research, King Saud University, Riyadh 11451, Saudi Arabia"}]},{"given":"Naif","family":"Alotaibi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad ibn Saud Islamic University (IMSIU), P. Box 90950, Riyadh 11623, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1093\/biomet\/84.3.641","article-title":"A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families","volume":"84","author":"Marshall","year":"1997","journal-title":"Biometrika"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1081\/STA-120003130","article-title":"Beta-normal distribution and its applications","volume":"31","author":"Eugene","year":"2002","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_3","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. 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