A novel size biased distribution: Regression model, INAR(1) process and Applications in Environmental and Medical Sciences

A novel size biased distribution: Regression model, INAR(1) process and Applications in Environmental and Medical Sciences | 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 A novel size biased distribution: Regression model, INAR(1) process and Applications in Environmental and Medical Sciences Na Elah, Peer Bilal Ahmad This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6998709/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 26 Nov, 2025 Read the published version in Environmental and Ecological Statistics → Version 1 posted 7 You are reading this latest preprint version Abstract A new statistical distribution, termed as Size-Biased Poisson Xgamma distribution, is introduced in this study to enhance modeling and interpretation of data arising from size-biased sampling schemes. In such settings, the probability of selecting a unit is proportional to its size, resulting in the overrepresentation of larger observations. In this work, we consider the specific case where the size of a unit is defined to be the observed value of the response variable, thereby focusing on size-biased distributions where selection is directly linked to outcome magnitude. We derive some structural features of the model and a thorough reliability analysis is also carried out. For parameter estimation, Maximum likelihood estimation (MLE) and method of moments (MoM) are used. As the MLE is not available in closed form, the likelihood function is maximized numerically and a simulation analysis based on MLE further validates the robustness of the model. Additionally, we extend the usefulness of the SBPXG model in predictive analytics by developing an associated regression model for applications with relevant covariate data. Furthermore, we present an INAR(1) process under the SBPXG model to handle count data scenarios. We investigate its special characteristics including conditional maximum likelihood (CML) method. By examining three different datasets, we show that the SBPXG model continuously performs better in terms of goodness-of-fit than other models. Additionally, the Vuong’s Likelihood Ratio Test is applied to formally compare non-nested models, and the results confirm that the SBPXG regression model offers a significantly better fit than conventional alternatives. The findings demonstrate the model’s flexibility and strong potential for analyzing size-biased and overdispersed count data, particularly in healthcare and environmental domains. Cardiovascular Count Data Length of Stay Poisson Xgamma Thunderstorm Likelihood Ratio Test Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 26 Nov, 2025 Read the published version in Environmental and Ecological Statistics → Version 1 posted Editorial decision: Accepted 20 Sep, 2025 Reviews received at journal 01 Sep, 2025 Reviewers agreed at journal 16 Jul, 2025 Reviewers invited by journal 13 Jul, 2025 Editor assigned by journal 08 Jul, 2025 Submission checks completed at journal 29 Jun, 2025 First submitted to journal 28 Jun, 2025 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. 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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-6998709","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":484760416,"identity":"85d8f3d8-5563-4bdb-be63-eee61b175518","order_by":0,"name":"Na Elah","email":"","orcid":"","institution":"Islamic University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Na","middleName":"","lastName":"Elah","suffix":""},{"id":484760418,"identity":"253e99c8-fee4-4001-869a-5ded297e3907","order_by":1,"name":"Peer Bilal Ahmad","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/klEQVRIiWNgGAWjYBACA2bmBoYHDAeAzAQGhg9Aio2doBbGBqBiiBbGGSAtzIS0MCBpYeYBCRHSYs7O2PghoeZONH977uHPNr+2yfMxMzB++JiDW4tlM2OzRMKxZ7kzzrxLMM7tu23YxszALDlzGx6HHWZskEhgO5zbcCPHIDm35zYjUAsbMy9+Lc0/Ev4dzp0P1HLYsue2PTFa2iQS2w7nbriRY9jM8ON2IkEtQL+0WST2Hc7deOZdMmNvw+3kNmbGZrx+Mec/fPjGh2+Hc+cdzz384cef27bz25sPfviIRwsSAEYKYxuIAYwpIgEoHv8Qq3gUjIJRMApGEgAAE5daeXwjPL0AAAAASUVORK5CYII=","orcid":"","institution":"Islamic University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Peer","middleName":"Bilal","lastName":"Ahmad","suffix":""}],"badges":[],"createdAt":"2025-06-28 15:23:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6998709/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6998709/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10651-025-00680-8","type":"published","date":"2025-11-26T15:58:32+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":97178593,"identity":"575118b2-8b39-403a-9247-11c402801d0d","added_by":"auto","created_at":"2025-12-01 16:11:13","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":767944,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptFolder.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6998709/v1_covered_a20e608b-651c-4eb7-aad2-ef41c772217e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A novel size biased distribution: Regression model, INAR(1) process and Applications in Environmental and Medical Sciences","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"environmental-and-ecological-statistics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"eest","sideBox":"Learn more about [Environmental and Ecological Statistics](http://link.springer.com/journal/10651)","snPcode":"10651","submissionUrl":"https://submission.nature.com/new-submission/10651/3","title":"Environmental and Ecological Statistics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Cardiovascular, Count Data, Length of Stay, Poisson Xgamma, Thunderstorm, Likelihood Ratio Test","lastPublishedDoi":"10.21203/rs.3.rs-6998709/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6998709/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eA new statistical distribution, termed as Size-Biased Poisson Xgamma distribution, is introduced in this study to enhance modeling and interpretation of data arising from size-biased sampling schemes. 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