det HτA H λ λHτ(A
m m m
+ n -21A ^1)^1 H
m J
(31)
times n2, converge jointly in distribution to the ordered solutions of the generalized
eigenvalue problem det[ W,m λ λVs;q r,m] = 0, where Ws,m ς ∑ Z1Y 2Y”Y” T,
(
V
s s,q-r,m
∑γ 2y.“y;
vk1
(32)
∖ 1(
(
m
Ib 1
Y
m
k= 1
m
Ik=1
** T
[cf. (23)], and the Yk** and X*k involved are independent s-variate and q-r-variate, respectively,
standard normal random vectors.
Note that the matrix Vs*, q-r,m in (32) differs from the matrix Vs*,m defined by (23) with r
replaced by s in that in the latter case the vectors X*k are (q-s) × 1 rather than (q-r) × 1.
If the null hypothesis (29) is false, then the matrix H can be written as
H R Rq rΓ1 + RrΓ2, where rank(Γ1) ^ s 1 ≥ 1, rank
(rʌ
γ1
2
V 2√
s.
(33)
Then again it follows straightforwardly from (15), (17) and (30) that
D m
H A. H →TτlRqlτrC(1)∑ XlXkτC(1)τRrrΓ,
k= 1
and
19