{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T12:16:52Z","timestamp":1777724212239,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:00:00Z","timestamp":1777420800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shuhrah Alghamdi","award":["(PNURSP2026R744)"],"award-info":[{"award-number":["(PNURSP2026R744)"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The QR distribution, recently introduced for modeling lifetime data under Type-II censoring, offers a flexible framework for survival and reliability analysis. This study provides the first comprehensive evaluation of multiple modern estimation techniques for the QR distribution under Type-II censoring. We systematically compare classical maximum likelihood estimation with stochastic gradient descent variants (Momentum and Adam), Bayesian approaches including Maximum A Posteriori estimation, Markov Chain Monte Carlo, and Variational Inference, as well as machine learning-integrated methods such as amortized neural network inference. Using both synthetic and the real Veterans\u2019 Administration Lung Cancer dataset, we evaluate these methods in terms of parameter estimation accuracy, computational efficiency, and convergence behavior. The results demonstrate the strengths of optimization-based, Bayesian, and neural approaches, highlighting their practical utility in handling complex censored survival data. This research validates the distribution\u2019s effectiveness in capturing survival dynamics, offering valuable insights for clinical applications and highlighting areas for methodological improvement.<\/jats:p>","DOI":"10.3390\/e28050502","type":"journal-article","created":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T10:34:42Z","timestamp":1777458882000},"page":"502","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient Estimation Methods for the QR Distribution with Type-II Censored Data: An Empirical Validation on Lung Cancer Prognosis"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4203-7798","authenticated-orcid":false,"given":"Qasim","family":"Ramzan","sequence":"first","affiliation":[{"name":"Department of Statistics, Government Graduate College Jauharabad, Khushab 41200, Pakistan"},{"name":"Department of Statistics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7431-5756","authenticated-orcid":false,"given":"Muhammad","family":"Amin","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-1149-8244","authenticated-orcid":false,"given":"Shuhrah","family":"Alghamdi","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Princess Nourah bint Abdulrahman University, Riyadh 11564, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6698-4933","authenticated-orcid":false,"given":"Randa","family":"Alharbi","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, University of Tabuk, Tabuk 47512, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,4,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3458","DOI":"10.1002\/qre.70028","article-title":"Bayesian and Classical Insights into a Novel Probability Model for Aircraft Windshield Failures: Bayesian and Classical Insights Into QR Distribution","volume":"41","author":"Ashraf","year":"2025","journal-title":"Qual. 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