{"title":"论多元单边问题似然比检验的无偏性","authors":"Yining Wang , Michael P. McDermott","doi":"10.1016/j.spl.2024.110231","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> be a random sample from a multivariate normal distribution with nonnegative mean <span><math><mi>μ</mi></math></span> and unknown covariance matrix <span><math><mi>Σ</mi></math></span>. The likelihood ratio test of <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>:</mo><mspace></mspace><mi>μ</mi><mo>=</mo><mi>0</mi></mrow></math></span> conditional on <span><math><mrow><mi>V</mi><mo>=</mo><mo>∑</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></math></span> is proven to be unbiased. Some related topics are also discussed.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the unbiasedness of the likelihood ratio test for the multivariate one-sided problem\",\"authors\":\"Yining Wang , Michael P. McDermott\",\"doi\":\"10.1016/j.spl.2024.110231\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> be a random sample from a multivariate normal distribution with nonnegative mean <span><math><mi>μ</mi></math></span> and unknown covariance matrix <span><math><mi>Σ</mi></math></span>. The likelihood ratio test of <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>:</mo><mspace></mspace><mi>μ</mi><mo>=</mo><mi>0</mi></mrow></math></span> conditional on <span><math><mrow><mi>V</mi><mo>=</mo><mo>∑</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></math></span> is proven to be unbiased. Some related topics are also discussed.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-08-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0167715224002001\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167715224002001","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the unbiasedness of the likelihood ratio test for the multivariate one-sided problem
Let be a random sample from a multivariate normal distribution with nonnegative mean and unknown covariance matrix . The likelihood ratio test of conditional on is proven to be unbiased. Some related topics are also discussed.
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