Bayesian Shrinkage Estimation of the Shape Parameter of the Power Distribution under Various Priors and Loss Functions

Bayesian Shrinkage Estimation of the Shape Parameter of the Power Distribution under Various Priors and Loss Functions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Bayesian Shrinkage Estimation of the Shape Parameter of the Power Distribution under Various Priors and Loss Functions Archana Jayaram, Jeevanand E S, Sowbhagya S Prabhu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8757190/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study develops and evaluates a suite of Bayesian shrinkage estimators for the shape parameter ($\alpha$) of the two-parameter Power distribution under a complete sampling scheme. The primary objective is to identify a robust estimation strategy that improves upon classical and standard Bayesian methods, particularly in scenarios with limited data. The methodological framework incorporates three distinct prior distributions (non-informative Uniform and Jeffreys; informative Exponential), five symmetric and asymmetric loss functions (Squared Error Loss Function (SELF), Weighted Squared Error Loss Function (WSELF), Kullback-Leibler Loss Function (KLF), Modified Quadratic Squared Error Loss Function (M/Q SELF), and Precautionary Loss Function (PLF)), and four shrinkage strategies (one with a constant factor and three modified Thompson-type adaptive estimators). Performance is rigorously assessed via a comprehensive Monte Carlo simulation study, comparing the proposed estimators against standard Maximum Likelihood and Bayes estimators in terms of absolute bias and mean squared error (MSE). The results consistently demonstrate the superiority of Bayesian methods over the Maximum Likelihood Estimator (MLE), especially for small sample sizes. Crucially, a modified shrinkage estimator based on the work of Mehta \& Srinivasan (1971) exhibits the lowest bias and MSE across nearly all considered scenarios, establishing its exceptional robustness and efficiency. The principal implication of this study is that the proposed adaptive shrinkage estimator offers a significant improvement in estimation accuracy for the Power distribution's shape parameter, presenting a valuable tool for practitioners in reliability engineering and lifetime data analysis. Applied Statistics Shrinkage type estimates Bayesian Estimation Power function distribution Symmetic and Asymmetric loss function Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8757190","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":583814945,"identity":"55c140c8-0650-4598-8625-b1092de3722d","order_by":0,"name":"Archana Jayaram","email":"","orcid":"https://orcid.org/0009-0008-4170-3956","institution":"Department of Mathematics and Statistics, St. Teresa’s College, Kerala, India","correspondingAuthor":false,"prefix":"","firstName":"Archana","middleName":"","lastName":"Jayaram","suffix":""},{"id":583814946,"identity":"445fe3b8-a2ce-4300-aa8c-5297d65b8ba5","order_by":1,"name":"Jeevanand E S","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAElEQVRIiWNgGAWjYHACMwYGHhibzQZIMDYeIEVLGkhLAxFa4IDtMJjCq8W8/fC2Bz9kbOQMjp89+Lmg7Lzd2vbDQFtqbKJxaZE5k1Zu2MOTZmxwJi9Zesa528nbziQCtRxLy23AoUWCIcdMgofncOLMhhwDad6228lmB4BaGBsO49bC/8ZM8g/P//qZ/W+Mf/O2nUs2O/+QgBaJHDNpHp4DCfwgBm/bATuzG4RskXhWJi3Dk2zYL/HGzJrnXHKC2Q2gLQn4/MKfvE3ybY+dPBt/jvFtnjI7e7Pz6Q8ffKixwakFDBh7EOxEsMoEfMrB4AeCaU9Q8SgYBaNgFIw4AACV0l0ESxoQAAAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-7102-4179","institution":"Department of Statistics and Data Science, Christ University, Karnataka, India","correspondingAuthor":true,"prefix":"","firstName":"Jeevanand","middleName":"E","lastName":"S","suffix":""},{"id":583814947,"identity":"5cf9349b-ad0b-4b7a-90f2-1dbb8ec99950","order_by":2,"name":"Sowbhagya S Prabhu","email":"","orcid":"https://orcid.org/0000-0002-1892-141X","institution":"Department of Statistics, Rajagiri College of Social Sciences, Kerala, India","correspondingAuthor":false,"prefix":"","firstName":"Sowbhagya","middleName":"S","lastName":"Prabhu","suffix":""}],"badges":[],"createdAt":"2026-02-01 15:11:19","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8757190/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8757190/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101731892,"identity":"a87884dc-21fb-4449-ad90-5378092e223f","added_by":"auto","created_at":"2026-02-03 06:18:39","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":260967,"visible":true,"origin":"","legend":"","description":"","filename":"paper1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8757190/v1_covered_ccea26e8-7bb1-4d71-9ce2-4d5240686ffb.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eBayesian Shrinkage Estimation of the Shape Parameter of the Power Distribution under Various Priors and Loss Functions\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Christ University, Bangalore","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Shrinkage type estimates, Bayesian Estimation, Power function distribution, Symmetic and Asymmetric loss function","lastPublishedDoi":"10.21203/rs.3.rs-8757190/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8757190/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study develops and evaluates a suite of Bayesian shrinkage estimators for the shape parameter ($\\alpha$) of the two-parameter Power distribution under a complete sampling scheme. The primary objective is to identify a robust estimation strategy that improves upon classical and standard Bayesian methods, particularly in scenarios with limited data. The methodological framework incorporates three distinct prior distributions (non-informative Uniform and Jeffreys; informative Exponential), five symmetric and asymmetric loss functions (Squared Error Loss Function (SELF), Weighted Squared Error Loss Function (WSELF), Kullback-Leibler Loss Function (KLF), Modified Quadratic Squared Error Loss Function (M/Q SELF), and Precautionary Loss Function (PLF)), and four shrinkage strategies (one with a constant factor and three modified Thompson-type adaptive estimators). Performance is rigorously assessed via a comprehensive Monte Carlo simulation study, comparing the proposed estimators against standard Maximum Likelihood and Bayes estimators in terms of absolute bias and mean squared error (MSE). 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