{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,3]],"date-time":"2026-01-03T14:50:42Z","timestamp":1767451842538,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,1]],"date-time":"2025-06-01T00:00:00Z","timestamp":1748736000000},"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","award":["QU-APC-2025"],"award-info":[{"award-number":["QU-APC-2025"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Probability models are instrumental in a wide range of applications by being able to accurately model real-world data. Over time, numerous probability models have been developed and applied in practical scenarios. This study introduces the AAP-X family of distributions\u2014a novel, flexible framework for continuous data analysis named after authors Aadil Ajaz and Parvaiz. The proposed family effectively accommodates both symmetric and asymmetric characteristics through its shape-controlling parameter, an essential feature for capturing diverse data patterns. A specific subclass of this family, termed the \u201cAAP Exponential\u201d (AAPEx) model is designed to address the inflexibility of classical exponential distributions by accommodating versatile hazard rate patterns, including increasing, decreasing and bathtub-shaped patterns. Several fundamental mathematical characteristics of the introduced family are derived. The model parameters are estimated using six frequentist estimation approaches, including maximum likelihood, Cramer\u2013von Mises, maximum product of spacing, ordinary least squares, weighted least squares and Anderson\u2013Darling estimation. Monte Carlo simulations demonstrate the finite-sample performance of these estimators, revealing that maximum likelihood estimation and maximum product of spacing estimation exhibit superior accuracy, with bias and mean squared error decreasing systematically as the sample sizes increases. The practical utility and symmetric\u2013asymmetric adaptability of the AAPEx model are validated through five real-world applications, with special emphasis on cancer survival times, COVID-19 mortality rates and reliability data. The findings indicate that the AAPEx model outperforms established competitors based on goodness-of-fit metrics such as the Akaike Information Criteria (AIC), Schwartz Information Criteria (SIC), Akaike Information Criteria Corrected (AICC), Hannan\u2013Quinn Information Criteria (HQIC), Anderson\u2013Darling (A*) test statistic, Cramer\u2013von Mises (W*) test statistic and the Kolmogorov\u2013Smirnov (KS) test statistic and its associated p-value. These results highlight the relevance of symmetry in real-life data modeling and establish the AAPEx family as a powerful tool for analyzing complex data structures in public health, engineering and epidemiology.<\/jats:p>","DOI":"10.3390\/sym17060863","type":"journal-article","created":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T03:57:18Z","timestamp":1748923038000},"page":"863","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Statistical Inference and Goodness-of-Fit Assessment Using the AAP-X Probability Framework with Symmetric and Asymmetric Properties: Applications to Medical and Reliability Data"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-3169-1311","authenticated-orcid":false,"given":"Aadil Ahmad","family":"Mir","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Kashmir, Srinagar 190006, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3079-5097","authenticated-orcid":false,"given":"A. A.","family":"Bhat","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Islamic University of Science and Technology, Awantipora 192122, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1809-1396","authenticated-orcid":false,"given":"S. P.","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Kashmir, Srinagar 190006, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1541-6239","authenticated-orcid":false,"given":"Badr S.","family":"Alnssyan","sequence":"additional","affiliation":[{"name":"Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8681-5774","authenticated-orcid":false,"given":"Abdelaziz","family":"Alsubie","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9410-9079","authenticated-orcid":false,"given":"Yashpal Singh","family":"Raghav","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1109\/24.229504","article-title":"Exponentiated Weibull family for analyzing bathtub failure-rate data","volume":"42","author":"Mudholkar","year":"1993","journal-title":"IEEE Trans. Reliab."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"6543","DOI":"10.1080\/03610926.2015.1130839","article-title":"A new method for generating distributions with an application to exponential distribution","volume":"46","author":"Mahdavi","year":"2017","journal-title":"Commun. 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