Using (A.2), one obtains
∂L |
I Hd = |
—2tr[ u 1(It (B0B)-1)]+ 2[U0 U1 (It (B,B)~1) U1 u] |
= |
—v tr[( v2 Jt + v2 Eτ ) In ] + v[u0[( V4 Jτ + v4 Eτ ) In ]u] (A.12) 2 ^1 ¾° 2 ¾ 1 ¾ ° | |
∂L |
= |
N(T — 1) N , 1 „0, 1 - 1 ʌʌ , 1 ʌ,, 1 . n vv2 vv2 + uu ( V4 jt in)u + vu (~et in)u = 0, 2¾° 2¾1 2 ¾1 2 ¾ ° |
jH0d = |
—2tr[ U 1(jt in)] + 2[u0 U 1(jt in) U 1u] (A.13) | |
nt 1 r„,z^ „ | ||
= |
— TTT + TΓ4 [u(JT 1N)u] = 0; 2¾ 1 2¾ 1 | |
∂L Ж |
jHod = |
—2tr[ U 1(¾2It (W + W0))] + ∣[u0 U 1(¾2It (W + W0)) U 1u] |
= |
-12 [u0(Eτ (W + W 0))u] + ⅛ [u0(J0τ (W + W 0))^] = D λ, (A.14) 2¾° 2¾1 |
where ¾° = ^0(Eτ In)u∕N(T — 1) and ¾2 = ^0(Jτ In)u∕N are the maximum likelihood
estimates of ¾° and σ2, and u is the maximum likelihood residual under the null hypothesis
Hd
H0 .
Therefore, the score vector under Hd is given by
0
0
(A.15)
ʌ
D λ
Using (A.5), the elements of the information matrix are given by
J11 = e [ — g(¾L)2 ] =2 tr[(((σι2jr + σJ2Eτ ) in ))] (A.16)
_ N l 1 T — 1 ʌ
= у M + ~°^~ ɔ;
J22 = E [ — @@22 ] =2 tr[(((¾Γ2 Jτ + ¾-2 Eτ ) In )( Jt In ))2] (A.17)
~∖~μ∕
_ NT2
= l¾f ’
j33 = E [ — @\2 ] =2tr[(((σ1 2 jT + σ!72ET) in)
= ¾4 tr ' ( ¾1 1J/ + ¾-4Eτ ) (2W2 + 2W 0W
• (¾°It (W + W0)))2] (A.18)
: ( ¾° + (T — 1))b,
4¾1 /
22