{"title":"色散参数的估计","authors":"W. A. Thompson","doi":"10.6028/JRES.066B.016","DOIUrl":null,"url":null,"abstract":"This paper deals with a topi c in multivariate a nalys is. Consider that a sample of size n+ 1 has been collected from a p-variate normal distribution having dispersion matrix (CT ii'). Let a;r/n denote the usual unbiased estimate of CTW. Further, let O< l<u be constants such that all cha racteristic roots of a matrix having t he Wishart distribution lie in t he interval [I, u] with probability I a . A t heorem of Roy, Bose, and Gnanadesikan [A nn . Math. Stat. 24. 513536 (1953); Biometrika 44, 399410 (1957)] may be stated as follows: The probability is 1a t hat every principal minor determinant of II (aii ') (CT jj') and of (CTii,)ul(a jj') is nonnegat ive. The previous result may be used to prove t he main theorem of the p resent paper. Theorem: T he probability is at least 1a t hat t he following system of re lat ions hold simultaneo us ly : ulai i ~CT 'i ~I-laii; j = l , ... , p and ICT ii,-}f (11-1 + 11)0 ii·1 ~7WI -1,1) (a iiai';') ! ~' j r= j'.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"46 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1962-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"14","resultStr":"{\"title\":\"Estimation of dispersion parameters\",\"authors\":\"W. A. Thompson\",\"doi\":\"10.6028/JRES.066B.016\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper deals with a topi c in multivariate a nalys is. Consider that a sample of size n+ 1 has been collected from a p-variate normal distribution having dispersion matrix (CT ii'). Let a;r/n denote the usual unbiased estimate of CTW. Further, let O< l<u be constants such that all cha racteristic roots of a matrix having t he Wishart distribution lie in t he interval [I, u] with probability I a . A t heorem of Roy, Bose, and Gnanadesikan [A nn . Math. Stat. 24. 513536 (1953); Biometrika 44, 399410 (1957)] may be stated as follows: The probability is 1a t hat every principal minor determinant of II (aii ') (CT jj') and of (CTii,)ul(a jj') is nonnegat ive. The previous result may be used to prove t he main theorem of the p resent paper. Theorem: T he probability is at least 1a t hat t he following system of re lat ions hold simultaneo us ly : ulai i ~CT 'i ~I-laii; j = l , ... , p and ICT ii,-}f (11-1 + 11)0 ii·1 ~7WI -1,1) (a iiai';') ! ~' j r= j'.\",\"PeriodicalId\":408709,\"journal\":{\"name\":\"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics\",\"volume\":\"46 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1962-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"14\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.6028/JRES.066B.016\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.6028/JRES.066B.016","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
This paper deals with a topi c in multivariate a nalys is. Consider that a sample of size n+ 1 has been collected from a p-variate normal distribution having dispersion matrix (CT ii'). Let a;r/n denote the usual unbiased estimate of CTW. Further, let O< l
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