Polynomial Classification EM for Reject Inference in P2P Credit Scoring: Semi-Supervised Learning Under Non-Random Selection

Polynomial Classification EM for Reject Inference in P2P Credit Scoring: Semi-Supervised Learning Under Non-Random Selection | 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 Polynomial Classification EM for Reject Inference in P2P Credit Scoring: Semi-Supervised Learning Under Non-Random Selection Abderrahim EL AMRANI, Badreddine BENYACOUB, Mohammed EL HAJ TIRARI, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9098066/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 Credit scoring models trained exclusively on accepted applicants suffer from sample selection bias, a problem amplified in peer-to-peer (P2P) lending where rejection rates exceed 90%. Reject inference methods attempt to correct this bias by incorporating information from rejected applicants, yet most approaches assume linear decision boundaries and rarely evaluate whether the underlying missing-data assumptions hold empirically. We propose LogisticCEMD2, a semi-supervised classification algorithm that integrates degree-2 polynomial feature expansion with ℓ1 regularization within the Classification Expectation-Maximization (CEM) framework. The polynomial extension captures pairwise feature interactions and quadratic effects while ℓ1 penalization performs automatic feature selection among the expanded feature set. We develop a mini-batch variant with proven convergence guarantees and O(Jd2(B + Inl)) computational complexity. We evaluate the method on LendingClub data comprising 1,266,782 accepted and 2,401,300 rejected loan applications (2007–2018). Empirical validation reveals that the selection mechanism is strongly non-random (classification AUC = 0.943), establishing that the Missing at Random assumption underlying CEM is substantially violated. All subsequent results therefore characterize CEM behavior under documented Missing Not at Random conditions. Across 25 independent runs comparing 12 classifiers, Friedman tests confirm significant performance differences (χ2 = 193.4, p < 0.001). With the standard hard classification step, LogisticCEMD2 achieves AUC 0.602 with particular strength in recall (0.656), but exhibits poor calibration (Brier = 0.304). An ablation study with Bonferronicorrected Wilcoxon signed-rank tests reveals that replacing the hard classification step with soft posterior weights (SoftEM) recovers both discrimination (AUC = 0.633, matching the supervised baseline) and calibration (Brier = 0.236; Hosmer- Lemeshow p = 0.083), demonstrating that the discrimination loss is caused by the binarization of posteriors rather than the CEM framework itself. Sensitivity analysis reveals performance degradation when unlabeled samples substantially exceed labeled samples. These results establish that polynomial CEM with soft assignment preserves both discrimination and calibration under MAR violation, while the hard C-step variant trades discrimination for sensitivity. Practical value depends on the label assignment strategy and the asymmetric cost structure typical of credit scoring. Credit scoring Reject inference Semi-supervised learning P2P lending Classification EM Polynomial features Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 16 Apr, 2026 Editor assigned by journal 16 Mar, 2026 Submission checks completed at journal 16 Mar, 2026 First submitted to journal 11 Mar, 2026 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-9098066","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":626104829,"identity":"4ed46a1d-36c2-4b08-8975-3550dc46c53e","order_by":0,"name":"Abderrahim EL AMRANI","email":"","orcid":"","institution":"Institut National de Statistique et d'Economie Appliquée","correspondingAuthor":false,"prefix":"","firstName":"Abderrahim","middleName":"EL","lastName":"AMRANI","suffix":""},{"id":626104832,"identity":"7b303e64-9868-4acb-b2f3-d08791d10551","order_by":1,"name":"Badreddine BENYACOUB","email":"","orcid":"","institution":"Institut National de Statistique et d'Economie Appliquée","correspondingAuthor":false,"prefix":"","firstName":"Badreddine","middleName":"","lastName":"BENYACOUB","suffix":""},{"id":626104833,"identity":"e447783a-ee58-45bb-81dc-5514a3c1ee6e","order_by":2,"name":"Mohammed EL HAJ TIRARI","email":"","orcid":"","institution":"Institut National de Statistique et d'Economie Appliquée","correspondingAuthor":false,"prefix":"","firstName":"Mohammed","middleName":"EL HAJ","lastName":"TIRARI","suffix":""},{"id":626104834,"identity":"d596c122-5165-4fca-a809-c5a83a18d8e5","order_by":3,"name":"Ismail El Moudden","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArklEQVRIiWNgGAWjYDACdhDBJiHHwJAAYhCjhRmixZhkLQyJDURr4WfmTnzMU2aR3s+eY8DwoewwYS2SzbybjXnOSeTO7HljwDjjHBFaDA7zbpPmbZPI3XAjx4CZt40ILfZQLekGIC1/idECNBmsJQGshZEYLRKHeTcbzjknYTiz51nBwZ5z6YS18Lf3bnzwpqxOnp89eeODH2XWhLWggAMkqh8Fo2AUjIJRgAsAAHGQM94rNKgWAAAAAElFTkSuQmCC","orcid":"","institution":"Old Dominion University","correspondingAuthor":true,"prefix":"","firstName":"Ismail","middleName":"El","lastName":"Moudden","suffix":""}],"badges":[],"createdAt":"2026-03-11 21:08:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9098066/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9098066/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107706958,"identity":"75674d66-c8ab-4d65-b8c7-e8cab296b815","added_by":"auto","created_at":"2026-04-24 09:19:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":382908,"visible":true,"origin":"","legend":"","description":"","filename":"manuscriptsubmission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9098066/v1_covered_2eaa1f0a-b5f2-4e2c-acba-ab1e86e18e68.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Polynomial Classification EM for Reject Inference in P2P Credit Scoring: Semi-Supervised Learning Under Non-Random Selection","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":"operations-research-forum","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Operations Research Forum](https://link.springer.com/journal/43069)","snPcode":"43069","submissionUrl":"https://submission.nature.com/new-submission/43069/3","title":"Operations Research Forum","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Credit scoring, Reject inference, Semi-supervised learning, P2P lending, Classification EM, Polynomial features","lastPublishedDoi":"10.21203/rs.3.rs-9098066/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9098066/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eCredit scoring models trained exclusively on accepted applicants suffer from sample selection bias, a problem amplified in peer-to-peer (P2P) lending where rejection rates exceed 90%. Reject inference methods attempt to correct this bias by incorporating information from rejected applicants, yet most approaches assume linear decision boundaries and rarely evaluate whether the underlying missing-data assumptions hold empirically. We propose LogisticCEMD2, a semi-supervised classification algorithm that integrates degree-2 polynomial feature expansion with ℓ1 regularization within the Classification Expectation-Maximization (CEM) framework. The polynomial extension captures pairwise feature interactions and quadratic effects while ℓ1 penalization performs automatic feature selection among the expanded feature set. We develop a mini-batch variant with proven convergence guarantees and O(Jd2(B + Inl)) computational complexity. We evaluate the method on LendingClub data comprising 1,266,782 accepted and 2,401,300 rejected loan applications (2007–2018). Empirical validation reveals that the selection mechanism is strongly non-random (classification AUC = 0.943), establishing that the Missing at Random assumption underlying CEM is substantially violated. All subsequent results therefore characterize CEM behavior under documented Missing Not at Random conditions. Across 25 independent runs comparing 12 classifiers, Friedman tests confirm significant performance differences (χ2 = 193.4, p \u0026lt; 0.001). With the standard hard classification step, LogisticCEMD2 achieves AUC 0.602 with particular strength in recall (0.656), but exhibits poor calibration (Brier = 0.304). An ablation study with Bonferronicorrected Wilcoxon signed-rank tests reveals that replacing the hard classification step with soft posterior weights (SoftEM) recovers both discrimination (AUC = 0.633, matching the supervised baseline) and calibration (Brier = 0.236; Hosmer- Lemeshow p = 0.083), demonstrating that the discrimination loss is caused by the binarization of posteriors rather than the CEM framework itself. Sensitivity analysis reveals performance degradation when unlabeled samples substantially exceed labeled samples. These results establish that polynomial CEM with soft assignment preserves both discrimination and calibration under MAR violation, while the hard C-step variant trades discrimination for sensitivity. Practical value depends on the label assignment strategy and the asymmetric cost structure typical of credit scoring.\u003c/p\u003e","manuscriptTitle":"Polynomial Classification EM for Reject Inference in P2P Credit Scoring: Semi-Supervised Learning Under Non-Random Selection","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-24 03:51:40","doi":"10.21203/rs.3.rs-9098066/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewersInvited","content":"","date":"2026-04-16T12:38:45+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-03-17T01:42:44+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-03-17T01:42:13+00:00","index":"","fulltext":""},{"type":"submitted","content":"Operations Research Forum","date":"2026-03-11T20:59:42+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"operations-research-forum","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Operations Research Forum](https://link.springer.com/journal/43069)","snPcode":"43069","submissionUrl":"https://submission.nature.com/new-submission/43069/3","title":"Operations Research Forum","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"27e5668e-cb69-4373-a3fe-596916ecbcdd","owner":[],"postedDate":"April 24th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-04-24T03:51:40+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-24 03:51:40","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9098066","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9098066","identity":"rs-9098066","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}