ABSTRACT

The problem of testing H0: S*=I against the alternatives H1: S*?I remains

invariant under the affine group G=(O(p), Ep) where O(p) is the multiplicative group of p×p orthogonal matrices, and Ep is the translation group, operating as

This induces in the space of the sufficient statistic the transformation

Lemma 8.1.1. A set of maximal invariants in the space of under the affine group G comprises the characteristic roots of B, that is, the roots of

Proof. Since det(g(B-?I)g')=det(B-?I), the roots of the equation det(B-?I)=0 are invariant under G. To see that they are maximal invariant suppose that det(B-?I)=0 and det(B*- ?I)=0 have the same roots, where B, B* are two symmetric positive definite matrices; we want to show that there exists a such that B*=gBg'. Since B, B* are symmetric positive definite matrices there exist orthogonal matrices g1, such that

where ? is a diagonal matrix whose elements are the roots of (8.10). Since we get

where Q.E.D. We shall denote the characteristic roots of B by R1,…, Rp. Similarly the

corresponding maximal invariants in the parametric space of (v, S*) under G are ?1, …, ?p, the roots of det(S*- ?I)=0. Under H0 all ?i=1, and under H1 at least one ?i?1. The likelihood ratio test criterion A in terms of the maximal invariants (R1,…, Rp) can be written as

The modified likelihood ratio test for testing H0: S=S0 rejects H0 when

where the constant C' depends on the size a of the test. Note that the modified likelihood ratio test is obtained from the corresponding likelihood ratio test by replacing the sample size N by N-1. Since e/N is constant, we do not change the constant term in the modified likelihood ratio test for the sake of convenience only.