{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:56:54Z","timestamp":1760241414080,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T00:00:00Z","timestamp":1520294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The first- and second-order optimum achievable exponents in the simple hypothesis testing problem are investigated. The optimum achievable exponent for type II error probability, under the constraint that the type I error probability is allowed asymptotically up to \r\n          \r\n            \r\n              \u03b5\r\n            \r\n          \r\n        , is called the \r\n          \r\n            \r\n              \u03b5\r\n            \r\n          \r\n        -optimum exponent. In this paper, we first give the second-order \r\n          \r\n            \r\n              \u03b5\r\n            \r\n          \r\n        -optimum exponent in the case where the null hypothesis and alternative hypothesis are a mixed memoryless source and a stationary memoryless source, respectively. We next generalize this setting to the case where the alternative hypothesis is also a mixed memoryless source. Secondly, we address the first-order \r\n          \r\n            \r\n              \u03b5\r\n            \r\n          \r\n        -optimum exponent in this setting. In addition, an extension of our results to the more general setting such as hypothesis testing with mixed general source and a relationship with the general compound hypothesis testing problem are also discussed.<\/jats:p>","DOI":"10.3390\/e20030174","type":"journal-article","created":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T12:16:27Z","timestamp":1520338587000},"page":"174","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["First- and Second-Order Hypothesis Testing for Mixed Memoryless Sources"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9744-3358","authenticated-orcid":false,"given":"Te Sun","family":"Han","sequence":"first","affiliation":[{"name":"National Institute of Information and Communications Technology (NICT), Tokyo 184-8795, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0799-615X","authenticated-orcid":false,"given":"Ryo","family":"Nomura","sequence":"additional","affiliation":[{"name":"School of Network and Information, Senshu University, Kanagawa 214-8580, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Han, T.S. 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Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5553","DOI":"10.1109\/TIT.2014.2339231","article-title":"Second-Order Slepian-Wolf Coding Theorems for Non-Mixed and Mixed Sources","volume":"60","author":"Nomura","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4395","DOI":"10.1109\/TIT.2016.2573310","article-title":"First- and Second-Order Coding Theorems for Mixed Memoryless Channels with General Mixture","volume":"62","author":"Yagi","year":"2016","journal-title":"IEEE Trans. Inf. Theory"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/3\/174\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:57:46Z","timestamp":1760194666000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/3\/174"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,6]]},"references-count":12,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,3]]}},"alternative-id":["e20030174"],"URL":"https:\/\/doi.org\/10.3390\/e20030174","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,3,6]]}}}