{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T02:08:05Z","timestamp":1740103685274,"version":"3.37.3"},"reference-count":25,"publisher":"Wiley","license":[{"start":{"date-parts":[[2023,3,29]],"date-time":"2023-03-29T00:00:00Z","timestamp":1680048000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah Bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2023R443","PSAU\/2023\/R\/1444"],"award-info":[{"award-number":["PNURSP2023R443","PSAU\/2023\/R\/1444"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009392","name":"Prince Sattam bin Abdulaziz University","doi-asserted-by":"publisher","award":["PNURSP2023R443","PSAU\/2023\/R\/1444"],"award-info":[{"award-number":["PNURSP2023R443","PSAU\/2023\/R\/1444"]}],"id":[{"id":"10.13039\/100009392","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2023,3,29]]},"abstract":"<jats:p>In this study, we use the idea of the hierarchical model (HM) to estimate an unknown parameter of the hierarchical Poisson-Gamma model using the E-Bayesian (E-B) theory. We propose the idea of hierarchical probability function instead of the traditional hierarchical prior density function. We aim to infer E-B estimates with respect to the conjugate Gamma prior distribution along with the E-posterior risks on the basis of different symmetric and asymmetric loss functions (LFs) under restricted and unrestricted parameter spaces using uniform hyperprior. Whereas, E-B estimators are compared with maximum likelihood estimators (MLEs) using mean squared error (MSE). Monte Carlo simulations are prosecuted to study the efficiency of E-B estimators empirically. It is shown that the LFs under a restricted parameter space dominate to estimate the parameter of the hierarchical Poisson-Gamma model. It is also found that the E-B estimators are more precise than MLEs, and Stein\u2019s LF has the least E-PR. Moreover, the application of outcomes to a real-life example has been made for analysis, comparison, and motivation.<\/jats:p>","DOI":"10.1155\/2023\/8767200","type":"journal-article","created":{"date-parts":[[2023,3,29]],"date-time":"2023-03-29T16:07:41Z","timestamp":1680106061000},"page":"1-19","source":"Crossref","is-referenced-by-count":0,"title":["E-Bayesian Estimation of Hierarchical Poisson-Gamma Model on the Basis of Restricted and Unrestricted Parameter Spaces"],"prefix":"10.1155","volume":"2023","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2329-1675","authenticated-orcid":true,"given":"Azeem","family":"Iqbal","sequence":"first","affiliation":[{"name":"Higher Education Department, Govt of the Punjab, Lahore 54000, Pakistan"},{"name":"Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2788-8946","authenticated-orcid":true,"given":"Laila A.","family":"Al-Essa","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3541-6220","authenticated-orcid":true,"given":"Muhammad Yousaf","family":"Shad","sequence":"additional","affiliation":[{"name":"Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan"},{"name":"Department of Mathematics, Namal University, Talagang Road, Mianwali 42250, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2255-1167","authenticated-orcid":true,"given":"Fuad S.","family":"Alduais","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11912, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3952-4341","authenticated-orcid":true,"given":"Mansour F.","family":"Yassen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11912, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Damietta, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5944-5544","authenticated-orcid":true,"given":"Muhammad Ahmad","family":"Raza","sequence":"additional","affiliation":[{"name":"Federal Urdu University of Arts Science and Technology, Islamabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","volume-title":"Introduction to the Theory of Statistics","author":"A. 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