An asymptotic formula and a series expansion for bivariate Normal tail probability

An asymptotic formula and a series expansion for bivariate Normal tail probability | 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 An asymptotic formula and a series expansion for bivariate Normal tail probability Siu-Kui Au This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3856455/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract This work presents a new asymptotic formula for the bivariate Normal tail probability. It only requires the larger threshold to grow indefinitely, but otherwise has no restriction on how the thresholds grow. The correlation parameter can change and possibly depend on the thresholds. The formula is applicable regardless of Salvage condition. Asymptotically, it reduces to Ruben’s formula and Hashorva’s formula under the corresponding conditions, and therefore can be considered a generalisation. Under a mild condition, it satisfies Plackett’s identity on the derivative with respect to correlation parameter. Motivated by the asymptotic formula, a series expansion in terms of the derivatives of univariate Mill’s ratio is also obtained for the exact tail probability, whose terms can be calculated recursively. Based on the series expansion, a simple procedure is developed for general numerical computation by suitable redefinition of parameters. Examples are presented to illustrate the theoretical findings. Bivariate Normal probability Hashorva’s formula Plackett’s identity Ruben’s formula Salvage condition Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 13 Jan, 2024 Editor assigned by journal 13 Jan, 2024 Submission checks completed at journal 13 Jan, 2024 First submitted to journal 12 Jan, 2024 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-3856455","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":266871430,"identity":"9983499b-6832-4842-89ca-513d68b88539","order_by":0,"name":"Siu-Kui Au","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVRIiWNgGAWjYBACNghlA6F4SNCSBlFNlBYoOEyCFj6J5GcPfradz7OXSGB88LaNwV7egZDDJNLMDXvbbhfzSCQwG85tY0jceICQFp4DZhK8bbcTeyQS2KR52xgSDBsIajn+TfJv2zmQFvbfQC32hLWw95gBDT8AtoUZqIVxPgEdIC1l0jLnkhN7zjxslpxzTiJxAyEt8s3s2yTflNkltrcnH/zwpszGXp6Qw5AAI0itBIPBAeK1wOwlwZZRMApGwSgYGQAA6+Y3lLCpD8QAAAAASUVORK5CYII=","orcid":"","institution":"Nanyang Technological University","correspondingAuthor":true,"prefix":"","firstName":"Siu-Kui","middleName":"","lastName":"Au","suffix":""}],"badges":[],"createdAt":"2024-01-12 09:44:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3856455/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3856455/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":49697603,"identity":"b348642d-10bc-4bcc-bd09-46952a07eeba","added_by":"auto","created_at":"2024-01-16 15:24:56","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":571149,"visible":true,"origin":"","legend":"","description":"","filename":"20240112bivariate.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3856455/v1_covered_aaac1b05-d4e6-4da1-a7f6-f2cbe2c1577e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An asymptotic formula and a series expansion for bivariate Normal tail probability","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":"statistics-and-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stco","sideBox":"Learn more about [Statistics and Computing](http://link.springer.com/journal/11222)","snPcode":"11222","submissionUrl":"https://submission.nature.com/new-submission/11222/3","title":"Statistics and Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Bivariate Normal probability, Hashorva’s formula, Plackett’s identity, Ruben’s formula, Salvage condition","lastPublishedDoi":"10.21203/rs.3.rs-3856455/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3856455/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis work presents a new asymptotic formula for the bivariate Normal tail probability. 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