{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T06:27:14Z","timestamp":1778912834134,"version":"3.51.4"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,27]],"date-time":"2022-06-27T00:00:00Z","timestamp":1656288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006261","name":"Taif University","doi-asserted-by":"publisher","award":["TURSP-2020\/316"],"award-info":[{"award-number":["TURSP-2020\/316"]}],"id":[{"id":"10.13039\/501100006261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We introduce here a new distribution called the power-modified Kies-exponential (PMKE) distribution and derive some of its mathematical properties. Its hazard function can be bathtub-shaped, increasing, or decreasing. Its parameters are estimated by seven classical methods. Further, Bayesian estimation, under square error, general entropy, and Linex loss functions are adopted to estimate the parameters. Simulation results are provided to investigate the behavior of these estimators. The estimation methods are sorted, based on partial and overall ranks, to determine the best estimation approach for the model parameters. The proposed distribution can be used to model a real-life turbocharger dataset, as compared with 24 extensions of the exponential distribution.<\/jats:p>","DOI":"10.3390\/e24070883","type":"journal-article","created":{"date-parts":[[2022,6,27]],"date-time":"2022-06-27T22:31:14Z","timestamp":1656369074000},"page":"883","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Power-Modified Kies-Exponential Distribution: Properties, Classical and Bayesian Inference with an Application to Engineering Data"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6723-6785","authenticated-orcid":false,"given":"Ahmed Z.","family":"Afify","sequence":"first","affiliation":[{"name":"Department of Statistics, Mathematics and Insurance, Benha University, Benha 13511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6767-4016","authenticated-orcid":false,"given":"Ahmed M.","family":"Gemeay","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nada M.","family":"Alfaer","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3052-6551","authenticated-orcid":false,"given":"Gauss M.","family":"Cordeiro","sequence":"additional","affiliation":[{"name":"Departamento de Estat\u00edstica, Universidade Federal de Pernambuco, Recife 50710-165, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2527-5938","authenticated-orcid":false,"given":"Eslam H.","family":"Hafez","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Helwan University, Helwan 11795, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Al-Babtain, A.A., Shakhatreh, M.K., Nassar, M., and Afify, A.Z. (2020). A New Modified Kies Family: Properties, Estimation Under Complete and Type-II Censored Samples, and Engineering Applications. Mathematics, 8.","DOI":"10.3390\/math8081345"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.csda.2013.02.026","article-title":"Power Lindley distribution and associated inference","volume":"64","author":"Ghitany","year":"2013","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2084236","DOI":"10.1155\/2016\/2084236","article-title":"On a power transformation of half-logistic distribution","volume":"2016","author":"Krishnarani","year":"2016","journal-title":"J. Probab. 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Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1080\/02664763.2018.1523376","article-title":"A power log-Dagum distribution: Estimation and applications","volume":"46","author":"Bakouch","year":"2019","journal-title":"J. Appl. Stat."},{"key":"ref_8","first-page":"429","article-title":"Power length-biased Suja distribution: Properties and application","volume":"12","author":"Alhyasat","year":"2019","journal-title":"Electron. J. Appl. Stat. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sobhi, A.L., and Mashail, M. (2020). The Inverse-Power Logistic-Exponential Distribution: Properties, Estimation Methods, and Application to Insurance Data. 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