Some Further Results in Simultaneous Confidence Interval Estimation

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 σ and σ₁σ⁻¹₂, where σ stands for the dispersion matrix of one p-variate and σ₁ and σ₂ for the dispersion matrices of two p-variate normal populations. Case II relates to a $(p + q)$-variate normal population (p 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 β(p × 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 σ and σ₁σ⁻¹₂ are given respectively by (3.1.3) and (3.2.8). Confidence bounds on the bilinear compounds of the regression matrix β are given by (4.7).