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This approach allows one to identify a parametric function class over which two given random variables may violate the non-parametric stochastic order with specific patterns. The parameter used to name the parametric function class also measures the ratio of dominance violation for the corresponding non-parametric stochastic orders. Our framework, expanding the domain of stochastic orders, covers the existing studies of almost stochastic dominance. This leads to intuitive explanations and simpler proofs of existing results and their extensions.<\/jats:p>","DOI":"10.1017\/jpr.2025.32","type":"journal-article","created":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T07:22:02Z","timestamp":1759303322000},"page":"1-31","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Integral stochastic orders with parametric classes of functions"],"prefix":"10.1017","volume":"63","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6176-044X","authenticated-orcid":false,"given":"Qi","family":"Feng","sequence":"first","affiliation":[{"name":"Purdue University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J. George","family":"Shanthikumar","sequence":"additional","affiliation":[{"name":"Purdue University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,10,1]]},"reference":[{"key":"S0021900225000324_ref3","doi-asserted-by":"publisher","DOI":"10.1111\/poms.13629"},{"key":"S0021900225000324_ref18","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.2016.2486"},{"key":"S0021900225000324_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-34675-5"},{"key":"S0021900225000324_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/jpr.2021.44"},{"key":"S0021900225000324_ref13","doi-asserted-by":"publisher","DOI":"10.3390\/jrfm15100438"},{"key":"S0021900225000324_ref5","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.2019.3406"},{"key":"S0021900225000324_ref4","first-page":"252","article-title":"A note on almost stochastic dominance","volume":"121","author":"Guo","year":"2013","journal-title":"J. Math. 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Wiley, Chichester."},{"key":"S0021900225000324_ref20","doi-asserted-by":"publisher","DOI":"10.1086\/705555"},{"key":"S0021900225000324_ref24","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.1120.1616"},{"key":"S0021900225000324_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2007.12.017"},{"key":"S0021900225000324_ref1","doi-asserted-by":"crossref","unstructured":"[1] Azmoodeh, E. and H\u00fcr, O. (2023). Multi-fractional stochastic dominance: Mathematical foundations. Available at arXiv:2307.08651.","DOI":"10.2139\/ssrn.4498983"},{"key":"S0021900225000324_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s10479-008-0470-7"},{"key":"S0021900225000324_ref2","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1037125858"},{"key":"S0021900225000324_ref26","doi-asserted-by":"publisher","DOI":"10.5018\/economics-ejournal.ja.2018-51"},{"key":"S0021900225000324_ref6","doi-asserted-by":"crossref","unstructured":"[6] Klar, B. and M\u00fcller, A. (2019). On consistency of the omega ratio with stochastic dominance rule. In Proceedings of the Innovations in Insurance, Risk- and Asset Management Conference: Innovations in Insurance, Risk- and Asset Management, eds K. Glau et al., pp. 367\u2013380. World Scientific.","DOI":"10.1142\/9789813272569_0014"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225000324","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:25:46Z","timestamp":1775525146000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225000324\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,1]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225000324"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.32","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,1]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}