{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T15:52:33Z","timestamp":1791215553351,"version":"4.1.0"},"reference-count":33,"publisher":"Oxford University Press (OUP)","license":[{"start":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T00:00:00Z","timestamp":1791158400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004490","name":"University of Waterloo","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004490","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>This paper focuses on learning from positive and unlabelled (PU) data, where only some positives are labelled and the rest are mixed with negatives. Classical exponential tilting models guarantee identifiability by imposing a linear structure, but they can be severely misspecified when the true relationships are nonlinear. We propose a generalized additive density-ratio framework that retains identifiability while allowing nonlinear and feature-specific effects. The approach comes with a practical fitting algorithm and supporting theory that enable estimation and inference for the mixture proportion and other quantities of interest. In simulations and analyses of real datasets, the proposed method matches the standard exponential tilting method when the linear model is correct and delivers clear gains when it is not. Overall, the framework strikes a useful balance between flexibility and interpretability for PU data and provides principled tools for estimation, prediction, and uncertainty assessment.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkag130","type":"journal-article","created":{"date-parts":[[2026,9,14]],"date-time":"2026-09-14T11:12:17Z","timestamp":1789384337000},"source":"Crossref","is-referenced-by-count":0,"title":["Modelling positive and unlabelled data with a generalized additive density ratio model"],"prefix":"10.1093","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5312-0204","authenticated-orcid":false,"given":"Peijun","family":"Sang","sequence":"first","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo , 200 University Ave West, Waterloo, ON N2L3G1 ,","place":["Canada"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yifan","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo , 200 University Ave West, Waterloo, ON N2L3G1 ,","place":["Canada"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9455-5466","authenticated-orcid":false,"given":"Qinglong","family":"Tian","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo , 200 University Ave West, Waterloo, ON N2L3G1 ,","place":["Canada"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Donglin","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, University of Michigan , 500 S State St, Ann Arbor, MI 48109 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pengfei","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo , 200 University Ave West, Waterloo, ON N2L3G1 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