{"title":"序列k决策问题的似然比检验","authors":"G. Lorden","doi":"10.1214/AOMS/1177692374","DOIUrl":null,"url":null,"abstract":"Sequential tests of separated hypotheses concerning the parameter θ of a Koopman-Darmois family are studied from the point of view of minimizing expected sample sizes pointwise in θ subject to error probability bounds. Sequential versions of the (generalized) likelihood ratio test are shown to exceed the minimum expected sample sizes by at most M log log α(-1) uniformly in θ, where α is the smallest error probability bound. The proof considers the likelihood ratio tests as ensembles of sequential probability ratio tests and compares them with alternative procedures by constructing alternative ensembles, applying a simple inequality of Wald and a new inequality of similar type. A heuristic approximation is given for the error probabilities of likelihood ratio tests, which provides an upper bound in the case of a normal mean.","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"22","resultStr":"{\"title\":\"Likelihood ratio tests for sequential k-decision problems\",\"authors\":\"G. Lorden\",\"doi\":\"10.1214/AOMS/1177692374\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Sequential tests of separated hypotheses concerning the parameter θ of a Koopman-Darmois family are studied from the point of view of minimizing expected sample sizes pointwise in θ subject to error probability bounds. Sequential versions of the (generalized) likelihood ratio test are shown to exceed the minimum expected sample sizes by at most M log log α(-1) uniformly in θ, where α is the smallest error probability bound. The proof considers the likelihood ratio tests as ensembles of sequential probability ratio tests and compares them with alternative procedures by constructing alternative ensembles, applying a simple inequality of Wald and a new inequality of similar type. A heuristic approximation is given for the error probabilities of likelihood ratio tests, which provides an upper bound in the case of a normal mean.\",\"PeriodicalId\":50764,\"journal\":{\"name\":\"Annals of Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1972-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"22\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Mathematical Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214/AOMS/1177692374\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/AOMS/1177692374","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Likelihood ratio tests for sequential k-decision problems
Sequential tests of separated hypotheses concerning the parameter θ of a Koopman-Darmois family are studied from the point of view of minimizing expected sample sizes pointwise in θ subject to error probability bounds. Sequential versions of the (generalized) likelihood ratio test are shown to exceed the minimum expected sample sizes by at most M log log α(-1) uniformly in θ, where α is the smallest error probability bound. The proof considers the likelihood ratio tests as ensembles of sequential probability ratio tests and compares them with alternative procedures by constructing alternative ensembles, applying a simple inequality of Wald and a new inequality of similar type. A heuristic approximation is given for the error probabilities of likelihood ratio tests, which provides an upper bound in the case of a normal mean.