{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T16:14:13Z","timestamp":1786119253410,"version":"build-2736575974"},"reference-count":31,"publisher":"Public Library of Science (PLoS)","issue":"8","license":[{"start":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T00:00:00Z","timestamp":1785974400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>This paper aiming at the formulation of a parsimonious two-parameter family of lifetime distributions obtained by a simple but flexible exponential-type transformation of an arbitrary baseline CDF (the Flexible Exponential-Type Family, FETF) and study in detail its Exponential-Type Weibull (ETW) special case. The ETW retains analytical tractability while permitting bathtub-shaped and unimodal hazard rate behavior frequently observed in epidemiology and reliability. To theoretically validate the model, we derive key distributional properties, including the probability density function, survival function, and hazard rate. We present maximum-likelihood estimation (MLE): score equations, observed Fisher information, and asymptotic confidence intervals. We address requirements that guarantee negative definiteness and provide explicit formulas for the Hessian (observed information). Monte Carlo simulations evaluating bias, mean squared error, and model selection frequency as well as application to COVID-19 data (n\u2009=\u2009106) from Mexican public records are used to empirically validate the performance of the proposed ETW distribution. The ETW consistently performs better than well-known competing models, such as the Weibull and other state-of-the-art distributions, across information requirements and goodness-of-fit metrics. For simulating non-monotonic hazard issues in epidemiology and reliability investigations, the ETW distribution provides a flexible and computationally easy tool.<\/jats:p>","DOI":"10.1371\/journal.pone.0331050","type":"journal-article","created":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T17:50:16Z","timestamp":1786038616000},"page":"e0331050","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["A flexible exponential type family for modeling non-monotonic hazard rates with application to mortality analysis: A COVID-19 case study"],"prefix":"10.1371","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-0365-5066","authenticated-orcid":true,"given":"Abdulrahman Obaid","family":"Alshammari","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hadi Obaid","family":"Alshammari","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hameed","family":"Ali","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-9384-1949","authenticated-orcid":true,"given":"Bilal","family":"Himmat","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,8,6]]},"reference":[{"issue":"5","key":"pone.0331050.ref001","doi-asserted-by":"crossref","first-page":"787","DOI":"10.3390\/sym13050787","article-title":"System of time fractional models for COVID-19: modeling, analysis and solutions","volume":"13","author":"O Iyiola","year":"2021","journal-title":"Symmetry"},{"issue":"2","key":"pone.0331050.ref002","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.3934\/mbe.2022057","article-title":"Some new results on bathtub-shaped hazard rate models","volume":"19","author":"M Kayid","year":"2022","journal-title":"Math Biosci Eng"},{"key":"pone.0331050.ref003","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1007\/s11222-023-10305-4","article-title":"Density estimation for toroidal data using semiparametric mixtures","volume":"33","author":"D Xu","year":"2023","journal-title":"Stat Comput"},{"issue":"2","key":"pone.0331050.ref004","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1080\/02331888.2024.2341792","article-title":"A new Marshall-Olkin lomax distribution with application using failure and insurance data","volume":"58","author":"M Ameeq","year":"2024","journal-title":"Statistics"},{"issue":"1","key":"pone.0331050.ref005","doi-asserted-by":"crossref","first-page":"33649","DOI":"10.1038\/s41598-025-14890-4","article-title":"A new exponential family of distributions with applications to engineering and medical data","volume":"15","author":"LP Sapkota","year":"2025","journal-title":"Sci Rep"},{"key":"pone.0331050.ref006","first-page":"123","article-title":"Modified Weibull distribution","volume":"11","author":"AM Sarhan","year":"2009","journal-title":"APPS Appl Sci"},{"issue":"2","key":"pone.0331050.ref007","doi-asserted-by":"crossref","first-page":"133","DOI":"10.17713\/ajs.v43i2.37","article-title":"The beta transmuted weibull distribution","volume":"43","author":"M Pal","year":"2014","journal-title":"Austrian J Stat"},{"issue":"1","key":"pone.0331050.ref008","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":"A Alzaatreh","year":"2013","journal-title":"METRON"},{"issue":"1","key":"pone.0331050.ref009","doi-asserted-by":"crossref","first-page":"53","DOI":"10.6339\/JDS.201401_12(1).0004","article-title":"The Weibull-G family of probability distributions","volume":"12","author":"M Bourguignon","year":"2021","journal-title":"J Data Sci"},{"issue":"24","key":"pone.0331050.ref010","doi-asserted-by":"crossref","first-page":"7326","DOI":"10.1080\/03610926.2014.980516","article-title":"The logistic-X family of distributions and its applications","volume":"45","author":"MH Tahir","year":"2016","journal-title":"Commun Stat - Theory Methods"},{"key":"pone.0331050.ref011","first-page":"106663","volume-title":"Exponentiated additive Weibull distribution. 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