Parametric Quantile Copula Regression for Bivariate Median Based Unit Rayleigh

Parametric Quantile Copula Regression for Bivariate Median Based Unit Rayleigh | 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 Parametric Quantile Copula Regression for Bivariate Median Based Unit Rayleigh Iman Mohammed Attia This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7827533/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 research introduces a comprehensive parametric copula regression framework, meticulously designed to model the joint behavior of two response variables, specifically conditioned on a chosen predictor variable. The innovative methodology effectively synthesizes parametric regression techniques for the marginal distributions with a copula function, which adeptly captures the intricate dependence structure between the response variables. Leveraging Sklar’s theorem, the formulation of the joint conditional density is achieved by expressing it as the product of the marginal densities and a copula density. Notably, the parameter of the copula is allowed to vary as a function of the covariate, enhancing the model's flexibility and adaptability. A Gumbel copula is utilized, which is particularly adept at modeling upper-tail dependence, and its parameter is intricately linked to the predictor through a smooth parametric function, ensuring a nuanced representation of relationships. For parameter estimation, the study employs the Inference Functions for Margins (IFM) method, which facilitates efficient and accurate estimation of model parameters. Additionally, a range of diagnostic tools based on the analysis of residuals are utilized to validate the adequacy of the model, ensuring that the fit is not only statistically sound but also meaningful in practical applications. The findings of this study reveal a significant trend: the dependence between the response variables tends to diminish as the predictor variable increases. This observation highlights the critical need to address associations that depend on covariates, a factor often overlooked in traditional analyses. Overall, this proposed framework stands out as a versatile and interpretable tool for analyzing complex multivariate relationships. Its potential for application spans various fields, making it a valuable resource in both academic research and professional practice. The framework offers possibilities for deeper insights into data that involves multiple interconnected response variables, reinforcing its relevance across diverse analytical contexts. Applied Mathematics Applied Statistics Copula parametric quantile regression Median Based Unit Rayleigh 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-7827533","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":527700730,"identity":"07cb70a0-360a-498a-8a5f-a97741bc78a4","order_by":0,"name":"Iman Mohammed Attia","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0002-7333-9713","institution":"Faculty of Graduate Studies for Statistical Research","correspondingAuthor":true,"prefix":"","firstName":"Iman","middleName":"Mohammed","lastName":"Attia","suffix":""}],"badges":[],"createdAt":"2025-10-10 13:31:01","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-7827533/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7827533/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93363351,"identity":"6b1b60e3-87e8-4132-97d2-dafbc7286193","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1820325,"visible":true,"origin":"","legend":"","description":"","filename":"QuantileCopularegressionalgorithm.docx","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/cd68ac6ad7695f73667009af.docx"},{"id":93363382,"identity":"8e92d64c-9054-435c-a651-e3a3c4dd0bf5","added_by":"auto","created_at":"2025-10-13 03:56:14","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":342,"visible":true,"origin":"","legend":"","description":"","filename":"rs7827533.json","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/97102905b5b4fe43ec4a5274.json"},{"id":93363315,"identity":"6ea02f27-eb6b-489f-a991-5f9d391a310a","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"xml","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":175089,"visible":true,"origin":"","legend":"","description":"","filename":"rs78275330enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/e068c871b35a51cf6846639f.xml"},{"id":93363381,"identity":"46bc6385-7af2-4571-b564-6dd2fc315ffb","added_by":"auto","created_at":"2025-10-13 03:56:13","extension":"png","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":52069,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/c14f3bac162f79039cf7d57a.png"},{"id":93363510,"identity":"c72fc33c-9b71-4ec9-9246-38be471a38bf","added_by":"auto","created_at":"2025-10-13 04:04:11","extension":"png","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":53180,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/5157db89061c68fc56fad99f.png"},{"id":93363347,"identity":"cf4a023e-13f2-4d7c-9f04-f616be4d0a9c","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":5,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":46103,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/b347da712cfe4271f80e8ca2.png"},{"id":93363310,"identity":"7e41f22e-c5cb-4314-a2e7-8c9f05f467a2","added_by":"auto","created_at":"2025-10-13 03:56:07","extension":"png","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":40700,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage12.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/9e8d37aa5fbfffb9a5d37006.png"},{"id":93363309,"identity":"519b8f03-aca9-4fff-8425-19e3da9077fe","added_by":"auto","created_at":"2025-10-13 03:56:07","extension":"png","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":41093,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage13.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/265a6c46f6834aef45f03055.png"},{"id":93363506,"identity":"315915f4-b5a8-4761-b01b-b4adea3bd637","added_by":"auto","created_at":"2025-10-13 04:04:08","extension":"png","order_by":8,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":39963,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage14.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/eae28d8d71032545c4f2d32f.png"},{"id":93363312,"identity":"d8a3eb39-618c-4684-b1d8-8628cefc76b7","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":42299,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage15.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/d7f88f1b5051267771c83f88.png"},{"id":93364157,"identity":"d8a270fd-44c3-46c0-a817-5342dc4b00c2","added_by":"auto","created_at":"2025-10-13 04:12:08","extension":"png","order_by":10,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":44703,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage16.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/3c1fc0b83c86acec90ded3a0.png"},{"id":93363327,"identity":"295522b0-46f3-4ee5-983c-299bd5ab768e","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":11,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":41245,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage17.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/e9304ee37612eb8f860ae9d9.png"},{"id":93363325,"identity":"7b61765a-7a55-45f2-b76a-d43958381d44","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":12,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":41080,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage18.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/4bab063f463f844615ecc46b.png"},{"id":93363313,"identity":"dbb28bfe-7a30-4abb-9fab-92348f94fff7","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":13,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":39990,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage19.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/69d2287b22a92d831eb2a405.png"},{"id":93363359,"identity":"d49b6a17-8a72-4eca-8d5f-e204a6ac26e1","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":14,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":256477,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/ef240e1da4ee694f21db35e9.png"},{"id":93363366,"identity":"0f42e5ee-6b33-4cb7-a994-48aa0569605c","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":15,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":41942,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage20.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/99fdbb6b6c8a5dfbca201433.png"},{"id":93363342,"identity":"dfe909a7-245d-4843-a7a0-5b688d7fa68d","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":16,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":50224,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage21.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/b0bc5f18c67686113fbe3c67.png"},{"id":93363319,"identity":"950d2239-4ec6-47be-ab85-afab364540c3","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":17,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":50387,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage22.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/7d120cceea6e1bd4c181fdad.png"},{"id":93363317,"identity":"16bf2fb6-9566-47b0-89fd-b28c55bc5c6e","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":18,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":53907,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage23.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/8faa30fd5defb8168639e852.png"},{"id":93363370,"identity":"5490d18d-0398-460f-9613-ea5848e5dbfb","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":19,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":53913,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage24.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/b951475313a6f992291ec67d.png"},{"id":93363326,"identity":"1c7f6a79-4e4e-440d-9e02-3449d9895d7a","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":20,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":47155,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage25.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/da98de34d6c0775f2772262b.png"},{"id":93363322,"identity":"4b8c4471-6f7f-4ea3-aeb0-b9da0e8526da","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":21,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":49666,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage26.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/afc65c17b6c7cc0310e9bc7c.png"},{"id":93363349,"identity":"7f221e9d-2885-4a4e-9171-5bb3245951d3","added_by":"auto","created_at":"2025-10-13 03:56:10","extension":"png","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":49845,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage27.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/0cf85887840a9cecab4ec575.png"},{"id":93363358,"identity":"9021f0ea-b3ce-4ebe-bbe8-fbf8f0fe95a9","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":23,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":47521,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage28.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/e1c28bf3a3426ea3f8ae9769.png"},{"id":93363335,"identity":"827e2d87-0d06-4f64-9f38-b68ecacb517e","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":24,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":54300,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage29.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/76bcf1fc19508de5c1026a0c.png"},{"id":93364158,"identity":"db2035a0-044b-463c-a98c-94fe14673223","added_by":"auto","created_at":"2025-10-13 04:12:08","extension":"png","order_by":25,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":19727,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/63e2bd4b04802137974d4e70.png"},{"id":93363337,"identity":"cebeec7d-1c91-4797-a4c6-36a10abcb32d","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":26,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":52195,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage30.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/cfbdaf37dbcaaa503fec8bd5.png"},{"id":93363364,"identity":"90ea7030-e937-4f41-a7d3-7c9cf34b3688","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":27,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":41760,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage31.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/f4476b740b8fba9a3ba80e2d.png"},{"id":93363352,"identity":"bb50d956-8545-4819-a60d-85a50455af0a","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":28,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":42883,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage32.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/2265f64b58273e98b72a8a25.png"},{"id":93363511,"identity":"698d6767-ae27-498c-ae19-4b02ed96c89b","added_by":"auto","created_at":"2025-10-13 04:04:11","extension":"png","order_by":29,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":30927,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage33.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/a01f14fbe09efa688b06129f.png"},{"id":93363504,"identity":"e4d34f61-d871-4577-945b-4a5e5210bfe7","added_by":"auto","created_at":"2025-10-13 04:04:08","extension":"png","order_by":30,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":31324,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage34.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/080be6e011601dac9d05c1a7.png"},{"id":93363512,"identity":"d71d91f9-bc22-4396-914f-52625d9bbfa4","added_by":"auto","created_at":"2025-10-13 04:04:11","extension":"png","order_by":31,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":28735,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/cda06a675b2bfd3f51467a3f.png"},{"id":93363356,"identity":"7009cea2-bf70-4dfd-89e0-94692f5b614b","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":32,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":40156,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/695c20145b0e61c216dcb1f9.png"},{"id":93363331,"identity":"8b00a793-12fd-49de-aa8a-7b4310984c28","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":33,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":82613,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/0528fb7e0b09353fbba0d743.png"},{"id":93363311,"identity":"bd854bc8-0ce0-4cdd-b022-b69365d7495d","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":34,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":25813,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/9d39296af5349541b180034a.png"},{"id":93363338,"identity":"8bb94391-928e-4574-b1c1-9927f893cab6","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":35,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":49651,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/8b3e1e147215c23e81fb51a3.png"},{"id":93363353,"identity":"f25d6f38-0615-43e7-9da5-166f80a26a34","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":36,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":47421,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/171aca7e49fce7d72e407d73.png"},{"id":93363355,"identity":"a7dbecc9-a7a7-43e6-96c3-88031c212b09","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":37,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":10504,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/565aee7e572713e84feef11f.png"},{"id":93363316,"identity":"aacfbcb2-c582-4656-8993-65af5d2d167e","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":38,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":21447,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/04d6a1c681e1e02744d63546.png"},{"id":93363330,"identity":"481fbe53-8df1-4dc6-bbb1-4ab74f0cf9cb","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":39,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20761,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/fa4655a43f6b6a0a6ee1fb6f.png"},{"id":93363314,"identity":"e760f82e-c59d-434f-b5c4-501df0775c77","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":40,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":18089,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage12.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/2fcfa13242cf4c30272fec35.png"},{"id":93363369,"identity":"0990de8e-c599-42c4-b48c-ba675e6267c8","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":41,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":18098,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage13.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/f708e37d2eab0ac573c27133.png"},{"id":93363373,"identity":"bddbc21e-eecc-46bd-87b5-5d75cfccf50c","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":42,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":19449,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage14.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/eaf002ebc35b38036a673da1.png"},{"id":93363329,"identity":"2994fef0-417c-4727-97ad-32e61253bf3c","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":43,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20399,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage15.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/5d11d7697e016ce53d55352f.png"},{"id":93363368,"identity":"3e0d5fd8-de7e-4f4a-99dd-7cc556a3d408","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":44,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20977,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage16.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/f8853e36a713085dc512f73e.png"},{"id":93363323,"identity":"f46a439c-e773-45e6-a835-30e80c17e564","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":45,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":18290,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage17.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/02e0cac92b29197206b91af0.png"},{"id":93363334,"identity":"8841dcba-b8a6-4403-a9e6-7d54aa374286","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":46,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":18053,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage18.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/3ba623f1e557b33954c934d2.png"},{"id":93363507,"identity":"459aad68-71cb-4edc-8d6c-c47ed56ae58c","added_by":"auto","created_at":"2025-10-13 04:04:08","extension":"png","order_by":47,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":19473,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage19.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/0fcaeedfb06e75f7ef33ee0e.png"},{"id":93363320,"identity":"70797de1-0d6f-4bf3-a899-11e6676b0d1c","added_by":"auto","created_at":"2025-10-13 03:56:08","extension":"png","order_by":48,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":34910,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/452f519e10893db481d67ca2.png"},{"id":93363376,"identity":"6ff495b7-aa4f-4c40-93fc-56dc5de111fd","added_by":"auto","created_at":"2025-10-13 03:56:12","extension":"png","order_by":49,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20337,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage20.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/1e2edebf4f10b142ef320bbc.png"},{"id":93363350,"identity":"19c213d3-94f6-4fe4-9497-45ff2c64be77","added_by":"auto","created_at":"2025-10-13 03:56:10","extension":"png","order_by":50,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20304,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage21.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/e26ea6109f297cc44343ed36.png"},{"id":93363339,"identity":"bfbd9cb5-9e82-4211-8bed-26d84cd3e3d9","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":51,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20915,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage22.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/95ad74b2606333654cb452e0.png"},{"id":93363377,"identity":"c591d824-1c76-4862-b5dd-e35a8a0ac2ce","added_by":"auto","created_at":"2025-10-13 03:56:12","extension":"png","order_by":52,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":28568,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage23.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/892eb58027f0333cb34ebec7.png"},{"id":93363372,"identity":"81c68bb7-520a-493c-ba90-90abbcebc975","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":53,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20644,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage24.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/0d847c7bc014fd347129055a.png"},{"id":93363345,"identity":"5bebc410-784d-40bb-adaa-f291808609ff","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":54,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":24773,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage25.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/cd818a434ba4314eae4fcae6.png"},{"id":93363375,"identity":"35f3492e-4d5e-44a0-9422-c712f11e55f4","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":55,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20642,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage26.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/c5d3f077d4914380c6b88c8a.png"},{"id":93363343,"identity":"94ec387d-61e2-4669-a9ab-a2bd69c11e5f","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":56,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20459,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage27.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/c9835b51ff6d26a731f25aef.png"},{"id":93363360,"identity":"454d4aa5-0616-408e-9a02-cc045acd5ed9","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":57,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":24919,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage28.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/9a012254332dfdba76e7fc33.png"},{"id":93363378,"identity":"9bcae2e4-4907-47fb-a7da-46f7b361272a","added_by":"auto","created_at":"2025-10-13 03:56:12","extension":"png","order_by":58,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":23425,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage29.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/2f9c9cbe52a1ae54183a33da.png"},{"id":93363348,"identity":"76001e0a-6245-423f-be4c-d79961cf94e4","added_by":"auto","created_at":"2025-10-13 03:56:10","extension":"png","order_by":59,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":10605,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/97d333c4a7433a09828e723f.png"},{"id":93363341,"identity":"64dcff0a-22af-4033-a581-485171096f77","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":60,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":21180,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage30.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/6963605104f50d206f835819.png"},{"id":93363340,"identity":"d87d49b2-bf6f-4c99-b5ff-376f5858c8f8","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":61,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":22843,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage31.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/176979058b11d154e48e7f24.png"},{"id":93363374,"identity":"8d1d6a22-29d7-42a0-9c31-a8f4bf275401","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":62,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":22993,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage32.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/a5fc465cc2d4a95783a12a8f.png"},{"id":93363380,"identity":"f6955aa0-ce36-45fb-8c7f-a733e6ffa7f6","added_by":"auto","created_at":"2025-10-13 03:56:12","extension":"png","order_by":63,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":15457,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage33.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/0c6cf496be3f6889dd47ecb3.png"},{"id":93363332,"identity":"77986318-06c4-4344-b61e-861ebab0d818","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":64,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":15401,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage34.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/8451505f3dc94e58a09f42a3.png"},{"id":93363354,"identity":"38f17d6c-7397-48d2-a434-04bc3bcda511","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":65,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":14171,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/dfc302d3286516c8163b5b4d.png"},{"id":93363379,"identity":"0a6c9f84-5e33-46dd-bd53-44c86bf68b90","added_by":"auto","created_at":"2025-10-13 03:56:12","extension":"png","order_by":66,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":19135,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/87a35990ed6455a39cfbd214.png"},{"id":93363344,"identity":"073a8c7c-e0fa-4011-9504-b1f7ccd918f1","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":67,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":21790,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/87b06a3ef4826e85a694f8a3.png"},{"id":93363346,"identity":"7aa3be2d-15a2-4b9a-95de-b0e52ce91086","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":68,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":13017,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/6cf2a4c863597b72fcbf02b0.png"},{"id":93363367,"identity":"2d7b3035-cd17-4550-8938-a5fb2e200770","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"png","order_by":69,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":19020,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/8ce77832d44731e7f47277ed.png"},{"id":93363336,"identity":"e1ede318-d3f5-4ff3-bbc7-9db6784abe32","added_by":"auto","created_at":"2025-10-13 03:56:09","extension":"png","order_by":70,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":21636,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/e69f8335e3ade3a020102282.png"},{"id":93363361,"identity":"46e10f21-fcac-40d1-9c55-ed082d3330a7","added_by":"auto","created_at":"2025-10-13 03:56:11","extension":"xml","order_by":71,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":171880,"visible":true,"origin":"","legend":"","description":"","filename":"rs78275330structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/9c8e7d5f2e78b53efbf43b04.xml"},{"id":93364160,"identity":"8442ffd9-ff87-4493-9b89-515f14b28758","added_by":"auto","created_at":"2025-10-13 04:12:11","extension":"html","order_by":72,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":200649,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1/10295b59893db5199c9dae64.html"},{"id":93364233,"identity":"437187b5-065a-4eb8-a940-4ec28d5a510f","added_by":"auto","created_at":"2025-10-13 04:20:09","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2104658,"visible":true,"origin":"","legend":"","description":"","filename":"QuantileCopularegressionalgorithm.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7827533/v1_covered_8a080a7c-a0ef-4112-8500-9eaae3219ebf.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eParametric Quantile Copula Regression for Bivariate Median Based Unit Rayleigh\u003c/strong\u003e\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Cairo University","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":"Copula, parametric quantile regression, Median Based Unit Rayleigh","lastPublishedDoi":"10.21203/rs.3.rs-7827533/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7827533/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis research introduces a comprehensive parametric copula regression framework, meticulously designed to model the joint behavior of two response variables, specifically conditioned on a chosen predictor variable. The innovative methodology effectively synthesizes parametric regression techniques for the marginal distributions with a copula function, which adeptly captures the intricate dependence structure between the response variables. Leveraging Sklar\u0026rsquo;s theorem, the formulation of the joint conditional density is achieved by expressing it as the product of the marginal densities and a copula density. Notably, the parameter of the copula is allowed to vary as a function of the covariate, enhancing the model's flexibility and adaptability. A Gumbel copula is utilized, which is particularly adept at modeling upper-tail dependence, and its parameter is intricately linked to the predictor through a smooth parametric function, ensuring a nuanced representation of relationships. For parameter estimation, the study employs the Inference Functions for Margins (IFM) method, which facilitates efficient and accurate estimation of model parameters. Additionally, a range of diagnostic tools based on the analysis of residuals are utilized to validate the adequacy of the model, ensuring that the fit is not only statistically sound but also meaningful in practical applications. The findings of this study reveal a significant trend: the dependence between the response variables tends to diminish as the predictor variable increases. This observation highlights the critical need to address associations that depend on covariates, a factor often overlooked in traditional analyses. Overall, this proposed framework stands out as a versatile and interpretable tool for analyzing complex multivariate relationships. Its potential for application spans various fields, making it a valuable resource in both academic research and professional practice. The framework offers possibilities for deeper insights into data that involves multiple interconnected response variables, reinforcing its relevance across diverse analytical contexts.\u003c/p\u003e","manuscriptTitle":"Parametric Quantile Copula Regression for Bivariate Median Based Unit Rayleigh","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-13 03:56:02","doi":"10.21203/rs.3.rs-7827533/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"c2407869-20e0-4f2a-8ac3-5d3c357e720b","owner":[],"postedDate":"October 13th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":56094331,"name":"Applied Mathematics"},{"id":56094332,"name":"Applied Statistics"}],"tags":[],"updatedAt":"2025-10-13T03:56:02+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-13 03:56:02","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7827533","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7827533","identity":"rs-7827533","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}