Consider the following generalization of the univariate linear model of Section 7.1. The variables Xi (i = 1,..., n) are given by Xi = ξi + Ui , where

(U1,..., Un) have a joint density which is spherical, that is, a function of n i=1 u2 i , say f (U1,..., Un) = q

U2 i



.

The parameter spaces  and ω and the hypothesis H are as in Section 7.1.

(i) The orthogonal transformation (7.1) reduces(X1,..., Xn)to canonical variables

(Y1,..., Yn) with Yi = ηi + Vi , where ηi = 0 for i = s + 1,..., n, H reduces to (7.3), and the V’s have joint density q(v1,...,vn).

(ii) In the canonical form of (i), the problem is invariant under the groups G1, G2, and G3 of Section 7.1, and the statistic W∗ given by (7.7) is maximal invariant.