ABSTRACT
From Lemma 8.1.1 a maximal invariant in the space of under G is R1,…, Rp, the characteristic roots of A. A corresponding maximal invariant in the parametric space is ?1,…, ?p, the characteristic roots of S*. In what follows in this section we write R, ? as diagonal matrices with diagonal elements R1,…, Rp, and ?1,…, ?p respectively. From (3.20) the probability ratio of the maximal invariant R is given by
where dµ(O) is the invariant probability measure on O(p). But
and (by (3.24))
where
with 0=a=1, d=tr(?- 1-I), q(i)(x)=(diq(x)/dxi). From (3.20), (8.23 and 8.24) the probability ratio in (8.233) is evaluated as (assuming q(2)(x)=0 for all x)
Using (8.233), the power function of any invariant level a test of H0: S*=I against H1: S*- I is positive definite, can be written as
If xq(1)(x)/q(x) is a decreasing function of x, the second term in (8.236) is maximized, by whenever tr R>constant and otherwise. Thus we get the following theorem.
