ABSTRACT
Let us now consider the distribution of the characteristic roots of A where A is distributed as Wp(I, n1). Since B is distributed as Wp(I, n2), B/n2? I almost surely as n2? 8 . Thus the roots of the equation det(A-?(B/n2))=0 converge almost surely to the roots of det(A-?I)=0. Let ?1>?2>···> ?p>0 be the ordered characteristic roots of A. To find the joint distribution of the ?i, it is sufficient to find the limit as n2? 8 of the probability density function of the roots of det(A-?(B/n2))=0. From (8.147), the probability density function of the roots (?1,…, ?p) of det(A-?(B/n2))=0 is given by
Since
we get
Thus the probability density function of the ordered characteristic roots ?1,…, ?p, of A is (with ? a diagonal matrix with diagonal elements ?1,…, ?p)
8.6.2. Multivariate Regression Model We now discuss a different formulation of the multivariate general linear hypothesis which is very appropriate for the analysis of design models. Let
Xa=(Xa1,…, Xap)', a=1,…, N, be independently distributed normal p-vectors with means
where za=(za1,…, Zas)', a=1,…, N are known vectors and ß=(ßij) is a p×s matrix of unknown elements ßij. As in the general formulation we shall assume that N-s=p,
and that the rank of the s×N matrix Z=(z1,…, zN) is s. Let ß=(ß1, ß2), where ß1, ß2 are submatrices of dimensions p×r and p×(s-r), respectively. We are interested in testing the null hypothesis
where ß2 and S are unknown. Here the dimension of pO is s s and that of p? is s-r.
The likelihood of the sample observations xa on Xa, a=1,…, N, is given by
