Some Further Results in Simultaneous Confidence Interval Estimation

Abstract

This paper is a follow-up of a previous paper [1], the full implications of some of the results there being brought out here in terms that are physically more meaningful. Two cases of simultaneous confidence bounds, I and II, are given, in each case with a confidence coefficient which is to be greater than or equal to a preassigned level. Case I relates to the characteristic roots of $\sigma$ and $\sigma_1\sigma^{-1}_2$, where $\sigma$ stands for the dispersion matrix of one $p$-variate and $\sigma_1$ and $\sigma_2$ for the dispersion matrices of two $p$-variate normal populations. Case II relates to a $(p + q)$-variate normal population $(p \leqq q)$, for which the matrix of regression of the $p$-set on the $q$-set is defined in a natural manner. This matrix is denoted by $\beta(p \times q)$ and simultaneous confidence bounds are given on all bilinear compounds of this matrix (with arbitrary coefficient vectors of unit modulus). Confidence bounds on the characteristic roots of $\sigma$ and $\sigma_1\sigma^{-1}_2$ are given respectively by (3.1.3) and (3.2.8). Confidence bounds on the bilinear compounds of the regression matrix $\beta$ are given by (4.7).