{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T10:41:09Z","timestamp":1770460869064,"version":"3.49.0"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T00:00:00Z","timestamp":1639699200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T00:00:00Z","timestamp":1639699200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007316","name":"Klaus Tschira Stiftung","doi-asserted-by":"publisher","award":["NA"],"award-info":[{"award-number":["NA"]}],"id":[{"id":"10.13039\/501100007316","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["257899354"],"award-info":[{"award-number":["257899354"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Karlsruher Institut f\u00fcr Technologie (KIT)"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2022,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Receiver operating characteristic (ROC) curves are used ubiquitously to evaluate scores, features, covariates or markers as potential predictors in binary problems. We characterize ROC curves from a probabilistic perspective and establish an equivalence between ROC curves and cumulative distribution functions (CDFs). These results support a subtle shift of paradigms in the statistical modelling of ROC curves, which we view as curve fitting. We propose the flexible two-parameter beta family for fitting CDFs to empirical ROC curves and derive the large sample distribution of minimum distance estimators in general parametric settings. In a range of empirical examples the beta family fits better than the classical binormal model, particularly under the vital constraint of the fitted curve being concave.<\/jats:p>","DOI":"10.1007\/s10994-021-06115-2","type":"journal-article","created":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T19:02:25Z","timestamp":1639767745000},"page":"2147-2159","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Receiver operating characteristic (ROC) curves: equivalences, beta model, and minimum distance estimation"],"prefix":"10.1007","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9397-3271","authenticated-orcid":false,"given":"Tilmann","family":"Gneiting","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Vogel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,12,17]]},"reference":[{"key":"6115_CR1","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1177\/0272989X9901900303","volume":"19","author":"R Etzioni","year":"1999","unstructured":"Etzioni, R., Pepe, M., Longton, G., Hu, C., & Goodman, G. (1999). Incorporating the time dimension in receiver operating characteristic curves: A case study of prostate cancer. Medical Decision Making, 19, 242\u2013251.","journal-title":"Medical Decision Making"},{"key":"6115_CR2","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1016\/j.patrec.2005.10.010","volume":"27","author":"T Fawcett","year":"2006","unstructured":"Fawcett, T. (2006). An introduction to ROC analysis. Pattern Recognition Letters, 27, 861\u2013874.","journal-title":"Pattern Recognition Letters"},{"key":"6115_CR3","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/s10994-007-5011-0","volume":"68","author":"T Fawcett","year":"2007","unstructured":"Fawcett, T., & Niculescu-Mizil, A. (2007). PAV and the ROC convex hull. Machine Learning, 68, 97\u2013106.","journal-title":"Machine Learning"},{"key":"6115_CR4","volume-title":"Encyclopedia of Machine Learning and Data Mining","author":"PA Flach","year":"2016","unstructured":"Flach, P. A. (2016). ROC analysis. In C. Sammut & G. I. Webb (Eds.), Encyclopedia of Machine Learning and Data Mining. Boston: Springer."},{"key":"6115_CR5","first-page":"25","volume":"24","author":"F Hsieh","year":"1996","unstructured":"Hsieh, F., & Turnbull, B. W. (1996). Nonparametric and semiparametric estimation of the receiver operating characteristic curve. Annals of Statistics, 24, 25\u201340.","journal-title":"Annals of Statistics"},{"key":"6115_CR6","doi-asserted-by":"publisher","DOI":"10.1201\/9781439800225","volume-title":"ROC Curves for Continuous Data","author":"DJ Krzanowski","year":"2009","unstructured":"Krzanowski, D. J., & Hand, D. (2009). ROC Curves for Continuous Data. Boca Raton: CRC Press."},{"key":"6115_CR7","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1080\/00029890.1984.11971392","volume":"91","author":"KM Levasseur","year":"1984","unstructured":"Levasseur, K. M. (1984). A probabilistic proof of the Weierstrass approximation theorem. American Mathematical Monthly, 91, 249\u2013250.","journal-title":"American Mathematical Monthly"},{"key":"6115_CR8","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1090\/S0002-9947-1984-0756045-0","volume":"286","author":"PW Millar","year":"1984","unstructured":"Millar, P. W. (1984). A general approach to the optimality of minimum distance estimators. Transactions of the American Mathematical Society, 286, 377\u2013418.","journal-title":"Transactions of the American Mathematical Society"},{"key":"6115_CR9","unstructured":"M\u00f6sching, A., & D\u00fcmbgen, L.\u00a0(2021). Estimation of a likelihood ratio ordered family of distributions \u2014 with a connection to total positivity. Preprint, arXiv:2007.11521v2."},{"key":"6115_CR10","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1016\/j.patrec.2019.10.004","volume":"128","author":"L Omar","year":"2019","unstructured":"Omar, L., & Ivrissimtzis, I. (2019). Using theoretical ROC curves for analysing machine learning binary classifiers. Pattern Recognition Letters, 128, 447\u2013451.","journal-title":"Pattern Recognition Letters"},{"key":"6115_CR11","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198509844.001.0001","volume-title":"The Statistical Evaluation of Medical Tests for Classification and Prediction","author":"MS Pepe","year":"2003","unstructured":"Pepe, M. S. (2003). The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford: Oxford University Press."},{"key":"6115_CR12","unstructured":"R Core Team. (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria"},{"key":"6115_CR13","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1186\/1471-2105-12-77","volume":"12","author":"X Robin","year":"2011","unstructured":"Robin, X., Turck, N., Hainard, A., Tiberti, N., Lisacek, F., Sanchez, J.-C., & M\u00fcller, M. (2011). pROC: An open-source package for R and S$$+$$ to analyze and compare ROC curves. BMC Bioinformatics, 12, 77.","journal-title":"BMC Bioinformatics"},{"key":"6115_CR14","doi-asserted-by":"publisher","first-page":"3940","DOI":"10.1093\/bioinformatics\/bti623","volume":"21","author":"T Sing","year":"2005","unstructured":"Sing, T., Sander, O., Beerenwinkel, N., & Lengauer, T. (2005). ROCR: Visualizing classifier performance in R. Bioinformatics, 21, 3940\u20133941.","journal-title":"Bioinformatics"},{"key":"6115_CR15","unstructured":"Vogel, P. (2019). Assessing Predictive Performance: From Precipitation Forecasts Over the Tropics to Receiver Operating Characteristic Curves and Back. PhD Thesis, Karlsruhe Institute of Technology, Faculty of Mathematics, available online at https:\/\/publikationen.bibliothek.kit.edu\/1000091649."},{"key":"6115_CR16","doi-asserted-by":"publisher","unstructured":"Vogel, P., & Jordan, A. I. (2021). Replication package for \u201cReceiver operating characteristic (ROC) curves: Equivalences, beta model, and minimum distance estimation\u201d (version v0.1.0) [data set]. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.4681331","DOI":"10.5281\/zenodo.4681331"},{"key":"6115_CR17","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1175\/WAF-D-17-0127.1","volume":"33","author":"P Vogel","year":"2018","unstructured":"Vogel, P., Knippertz, P., Fink, A. H., Schlueter, A., & Gneiting, T. (2018). Skill of global raw and postprocessed ensemble predictions of rainfall over northern tropical Africa. Weather and Forecasting, 33, 369\u2013388.","journal-title":"Weather and Forecasting"},{"key":"6115_CR18","doi-asserted-by":"publisher","DOI":"10.1002\/9780470906514","volume-title":"Statistical Methods in Diagnostic Medicine","author":"X-H Zhou","year":"2011","unstructured":"Zhou, X.-H., Obuchowski, N. A., & McClish, D. K. (2011). Statistical Methods in Diagnostic Medicine (2nd ed.). Hoboken: Wiley.","edition":"2"},{"key":"6115_CR19","doi-asserted-by":"publisher","first-page":"1259","DOI":"10.1002\/sim.1723","volume":"23","author":"KH Zou","year":"2004","unstructured":"Zou, K. H., Wells, W. M., III., Kikinis, R., & Warfield, S. K. (2004). Three validation metrics for automated probabilistic image segmentation of brain tumours. Statistics in Medicine, 23, 1259\u20131282.","journal-title":"Statistics in Medicine"}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-021-06115-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-021-06115-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-021-06115-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T12:23:17Z","timestamp":1699964597000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-021-06115-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,17]]},"references-count":19,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["6115"],"URL":"https:\/\/doi.org\/10.1007\/s10994-021-06115-2","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,17]]},"assertion":[{"value":"29 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 October 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 October 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 December 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}