Absolutely Continuous Bivariate Modified Weibull Distribution: Properties, Estimation

Absolutely Continuous Bivariate Modified Weibull Distribution: Properties, Estimation | 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 Absolutely Continuous Bivariate Modified Weibull Distribution: Properties, Estimation Sanjay Kumar This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7208393/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 The Block and Basu bivariate exponential (BBBE) distribution is one of the most popular and widely used absolutely continuous bivariate distributions. Kundu and Gupta \cite{kundu2010class} obtained the Block-Basu bivariate Weibull (BBBW) distribution. Extensive work has been done on the BBBW model over the past several decades. Interestingly, it is observed that the BBBW model can be extended to the modified Weibull model also. We call this new model as the Block and Basu bivariate modified Weibull (BBBMW) distribution. We consider the properties of the BBBMW distribution and provide the associated copula function. The BBBMW model has five unknown parameters and the maximum likelihood estimators (MLEs) cannot be obtained in closed form. To compute the MLEs directly, one needs to solve a five-dimensional optimization problem. We propose to use the EM algorithm for computing the MLEs of the unknown parameters. The proposed EM algorithm can be carried out by solving a two-dimensional optimization problem at each EM step. An extensive simulation is carried out, which demonstrates that the proposed EM algorithm performs quite well. A real data set is analyzed for illustrative purposes. Modified Weibull distribution Block-Basu bivariate distribution Mixture distribution Copula function Maximum Likelihood EM Algorithm Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 23 Nov, 2025 Reviews received at journal 23 Nov, 2025 Reviewers agreed at journal 23 Nov, 2025 Reviewers invited by journal 11 Aug, 2025 Editor assigned by journal 07 Aug, 2025 Submission checks completed at journal 07 Aug, 2025 First submitted to journal 24 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. 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-7208393","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":498698550,"identity":"fcd42166-ddfa-4c22-bdbf-498159c8d826","order_by":0,"name":"Sanjay Kumar","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABA0lEQVRIiWNgGAWjYBACCQTzAAjbAAnGxgOkaEkDaWkgVgtY22G4bpxAckZ24ucKhjp5c8bDz6RunDlvt7b9MNCWGptoXFqkJXI3S55hOGy4s+GYmXTOjdvJ284kArUcS8ttwKFFTiJ3g2QDwwHGDQcOALV8uJ1sdgCohbHhMD4tm382MNTZbzhw/BtQy7lks/MP8WsBOmwb0BbmxA0HzoAcdsDO7AYBWyR73m6zbDA4nAzUUmydcyY5wewG0JYEPH6ROJ67+WZDRZ3thhvHN97OOWZnb3Y+/eGDDzU2OLVAgAFI8wEWUCQlglUm4FUOA/wNzB+AlD1RikfBKBgFo2BEAQBOvmyRHyCDbQAAAABJRU5ErkJggg==","orcid":"","institution":"Indian Institute of Technology Kanpur","correspondingAuthor":true,"prefix":"","firstName":"Sanjay","middleName":"","lastName":"Kumar","suffix":""}],"badges":[],"createdAt":"2025-07-24 19:38:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7208393/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7208393/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89344459,"identity":"a8632af5-86a3-46d7-97c3-a95ecec5e4ea","added_by":"auto","created_at":"2025-08-19 04:18:47","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":475799,"visible":true,"origin":"","legend":"","description":"","filename":"SubmissionJSTA.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7208393/v1_covered_493fb745-6c24-40bf-b606-310ff733bfd4.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Absolutely Continuous Bivariate Modified Weibull Distribution: Properties, Estimation","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":"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":"Modified Weibull distribution, Block-Basu bivariate distribution, Mixture distribution, Copula function, Maximum Likelihood, EM Algorithm","lastPublishedDoi":"10.21203/rs.3.rs-7208393/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7208393/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The Block and Basu bivariate exponential (BBBE) distribution is one of the most popular and widely used absolutely continuous bivariate distributions. Kundu and Gupta \\cite{kundu2010class} obtained the Block-Basu bivariate Weibull (BBBW) distribution. Extensive work has been done on the BBBW model over the past several decades. Interestingly, it is observed that the BBBW model can be extended to the modified Weibull model also. We call this new model as the Block and Basu bivariate modified Weibull (BBBMW) distribution. We consider the properties of the BBBMW distribution and provide the associated copula function. The BBBMW model has five unknown parameters and the maximum likelihood estimators (MLEs) cannot be obtained in closed form. To compute the MLEs directly, one needs to solve a five-dimensional optimization problem. We propose to use the EM algorithm for computing the MLEs of the unknown parameters. The proposed EM algorithm can be carried out by solving a two-dimensional optimization problem at each EM step. An extensive simulation is carried out, which demonstrates that the proposed EM algorithm performs quite well. A real data set is analyzed for illustrative purposes.","manuscriptTitle":"Absolutely Continuous Bivariate Modified Weibull Distribution: Properties, Estimation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-19 03:46:41","doi":"10.21203/rs.3.rs-7208393/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-11-24T04:10:13+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-24T04:08:26+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"50007491096122979649793062160526183259","date":"2025-11-24T04:02:40+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-08-11T10:17:53+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-07T13:29:03+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-08-07T05:16:20+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Statistical Theory and Applications","date":"2025-07-24T19:26:38+00:00","index":"","fulltext":""}],"status":"published","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}}],"origin":"","ownerIdentity":"bba1eb35-a9d3-46db-a576-9f2c3fb5952f","owner":[],"postedDate":"August 19th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2025-12-19T12:54:13+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-19 03:46:41","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7208393","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7208393","identity":"rs-7208393","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}