ABSTRACT
Similarly a maximal invariant in the space of W under O(N-s) is
The problem remains invariant under the full linear group Gl(p) (multiplicative
group of p×p nonsingular matrices) of transformation g transforming Z to gZ, W to gW. The corresponding induced transformation in the space of (A, B) is given by (A, B)? (gAg', gBg'). By Exercise 7 the roots of det(A-?B)=0 (the characteristic roots of AB-1) are maximal invariant in the space of (A, B) under Gl(p). Let R1,…, Rp denote the roots of det(A-?B)=0. A corresponding maximal invariant in the parametric space is (?1,…, ?p), the characteristic roots of ? ? 'S
- 1 where ? =E(Z'). The test statistic U in (8.124) can be written as
Anderson (1958) called this statistic Up,r,N-s. Some other invariant tests are also proposed for this problem. They are as follows. In all cases the constant c will depend on the level of significance a of the test.
