Sample Size Determination for Skewed and Heavy-tailed Distributions

Sample Size Determination for Skewed and Heavy-tailed Distributions | 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 Sample Size Determination for Skewed and Heavy-tailed Distributions Jordan Slessor, Rui Hu, Zhichun Zhai This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7246683/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract In this article, we propose methods to determine sufficient sample sizes for applying the classical central limit theorem to skewed and heavy-tailed distributions. In doing so, we review the properties of an α−stable distribution and its domain of attraction. Then, we apply the general Edgeworth expansion for regularly varying distributions to t−distributions with degree freedom at least three. Motivated by the observed results, we propose a mathematical formula for determining enough sample sizes. The formula is valid for distributions with at least the fourth moment. Then, we propose an algorithm to apply this formula for a data set from general distributions. For distributions with infinite/undefined skew-ness/kurtosis, such as some heavy-tailed distributions, we could use Monte-Carlo simulation method to determine sample size. As an example, we propose an empirical method to determine the sample sizes for Pareto distributions. Both the algorithm and the empirical method are tested on simulated data. Central Limit Theorem α−distribution Regularly Varying Distribution Pareto Distribution Sample Size Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 30 Oct, 2025 Reviewers agreed at journal 05 Oct, 2025 Reviewers agreed at journal 05 Oct, 2025 Reviewers invited by journal 05 Oct, 2025 Editor assigned by journal 07 Aug, 2025 Submission checks completed at journal 07 Aug, 2025 First submitted to journal 29 Jul, 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-7246683","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":525087903,"identity":"6d9c1563-58c0-4a55-8a79-9085d23e1e72","order_by":0,"name":"Jordan Slessor","email":"","orcid":"","institution":"University of Alberta Edmonton","correspondingAuthor":false,"prefix":"","firstName":"Jordan","middleName":"","lastName":"Slessor","suffix":""},{"id":525087905,"identity":"d1ee3995-dab7-4ee0-b1a3-534bf4c923d4","order_by":1,"name":"Rui Hu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAr0lEQVRIiWNgGAWjYBACAwYG5gcfSNXCZjiDVC0M0jwkaTGXyH1gbJtjJ88vkfzsAUONHWEtljPSDR7nbks2nDkjzdyA4VgyEQ67kcZgnLuNOcHgdoKZBGMDM3FapC231SfY307/BtRST6QWxm2HEwykc0C2HCZCy5lnbIa9244bzrj/pkwi4dhxIrQcT2N+8HNbtTx/z/FtEh9qqglrQQUJpGoYBaNgFIyCUYAdAABlaTSKZJrCLgAAAABJRU5ErkJggg==","orcid":"","institution":"MacEwan University Edmonton","correspondingAuthor":true,"prefix":"","firstName":"Rui","middleName":"","lastName":"Hu","suffix":""},{"id":525087906,"identity":"b5b8ffff-dbcf-4d6c-a653-b063b63939dd","order_by":2,"name":"Zhichun Zhai","email":"","orcid":"","institution":"MacEwan University Edmonton","correspondingAuthor":false,"prefix":"","firstName":"Zhichun","middleName":"","lastName":"Zhai","suffix":""}],"badges":[],"createdAt":"2025-07-29 21:08:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7246683/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7246683/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93698255,"identity":"080cfa48-df87-40c7-92b1-671a4616b540","added_by":"auto","created_at":"2025-10-16 15:08:13","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4418,"visible":true,"origin":"","legend":"","description":"","filename":"7363ace9f8eb493ea51140330a500cc3.json","url":"https://assets-eu.researchsquare.com/files/rs-7246683/v1/684f567496df0229d3cde1c1.json"},{"id":93699213,"identity":"7bad05ee-8274-4c71-8c12-47388f25fc20","added_by":"auto","created_at":"2025-10-16 15:16:13","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":271868,"visible":true,"origin":"","legend":"","description":"","filename":"SampleSizeSlessorHuZhai.tex.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7246683/v1_covered_8fa1f27e-9a68-4a69-8984-fce6b5a83acc.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Sample Size Determination for Skewed and Heavy-tailed Distributions","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-statistical-theory-and-applications","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Journal of Statistical Theory and Applications](https://link.springer.com/journal/44199)","snPcode":"44199","submissionUrl":"https://submission.nature.com/new-submission/44199/3","title":"Journal of Statistical Theory and Applications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Open","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Central Limit Theorem, α−distribution, Regularly Varying Distribution, Pareto Distribution, Sample Size","lastPublishedDoi":"10.21203/rs.3.rs-7246683/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7246683/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this article, we propose methods to determine sufficient sample sizes for applying the classical central limit theorem to skewed and heavy-tailed distributions. 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