Notes on multivariate confidence bounds
Notes on multivariate confidence bounds
复制标题
关于多元置信界限的注释
DOI：
10.1007/bf01682326
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发表时间：
1960
影响因子：
1
通讯作者：
M. Siotani
中科院分区：
文献类型：
--
作者：
M. Siotani
1. 引言和总结。设x为p变量正态分布的观测值，均值向量为/2~，协方差矩阵为A，即N (Fz， A)， I= 1,2，…n;I = 1,2，…让我们使用下列常用的符号和定义：= ~ x / n。对于第i个样本的均值向量，分别为~ s和~ s的加权大均值向量=~ n~ n和/~= n~/l~ n，对于k个样本的“内”协方差矩阵。1953年，SN Roy和RC Bose[2]同时得到了k均值向量/2~ s与其加权大均值/~之差的所有任意双线性化合物的置信限，即对于所有/A，所有非零p维标量向量a和所有b~‘s服从~ b~= l,(1)~ b~ n~/~ a‘（X~—~）—[(k—1)c~ a’ la]‘1~ i= l<=~ b~ n~ a‘([~—~))<= b~ n~ a‘（[~—~]）§[(k—1)c~ a’ la]~”，其中c~是c:[l *—cL[= 0]中p次行列式方程的最大根的loa百分比点，其中l *是’between’协方差矩阵，定义为/c (k-1) l *= E n~(~—~ ~+~)(~-~-/~+/~ +)’。我=
1. Introduction and summary. Let x. be an observation from p-variate normal distribution with the mean vector/2~ and the covariance matrix A, ie, N (Fz, A), I= 1, 2,..., n,; i= 1, 2,..., k. Let us use the following usual notations and definitions:~.,=~ x./n. for the mean vector for the i-th sample,=~ n~ n~ and/~= n~/l~ n~, for the weighted grand mean vectors of~'s and~'s respectively, and for the pooled'within'covariance matrix of the k samples. In 1953, SN Roy and RC Bose [2] obtained the simultaneous confidence bounds on all arbitrary double linear compounds of the difference between k mean vectors/2~'s and their weighted grand mean/~, that is, for all/A's, all non-null p-dimensional scalar vector a's and all b~'s subject to~ b~= l,(1)~ b~ n~/~ a'(X~--~)--[(k--1) c~ a'La]'1~ i= l<=~ b~ n~ a'([~-~)<= b~ n~/~ a'(~-Ji) § [(k-1) c~ a'La]~" where c~ is the lOOa per cent point of the largest root of the p-th degree determinantal equation in c:[L*--cL [= O, where L* is the'between'covariance matrix, which is defined by/c (k-1) L*= E n~(~-~-~+~)(~-~-/~+/~)'. i= l