When ^ is equivariant, IF(X,Y; ^; Fθ) = σIF(u; ^5Φ). Hence V = σ 4C
with C a partitioned matrix with blocks
C11 = 2 Ek [vech( XX ' )(vech( XX ' )) ' ]
+(Bιι - 2)Ek[vech(XX')]Ek[vech(XX')]',
C22 = B 22 Ek ( XX ' ),
C33 = B33 ,
C13 = B13 Ek [vech( XX ' )] = C 31,
C12 = C21 = 0, C 23 = C3 2 = 0,
and all Bij as in Section 3. Again, V does not depend on the choice of the
estimator of the location parameter β . Replacing EK with sample averages
yields an estimate C of C. For a given θ = (β' , ^) ', let Ui = (Yi — X,i∣3')/σ
and
1 n / (U2 — 1)vech(XiX') ∖ / N^1 ∖
N = - Σ ( u 3 — 3 u i ) Xi = N^2 ∙
n i=1 ∖ U4 — 5 U 2 + 2 / ∖ N3 /
Then T takes the form
(ʌ » ʌ 1 1 ʌ ʌ » ʌ nn ʌ ʌ , ʌ OO ʌ ʌ , ʌ to ʌ ∖
N1 C11TV1 + N2C22IN2 + N3C33N3 + 2N1C13N3∖ ,
where Cij is the (i,j)-th block of (C+.
If θ is the ML estimator and the first column of X is a vector of ones, then
the first element of C11 is zero, B13 =0andB33 = 24. Let p = k(k + 1)/2,
with k =dim(/), and
L = 0p-1×1 Ip-1 ,
Li = L vech( XiX'),
1n
L = - ∑ Li∙
n
i=1
Then the IM test statistic with ML estimator can be written as (Hall, -987)
n n -1 n
T = 2∑(U2 — 1)Li{∑(Li — L)(Li — L)T ∑(U2 — 1)Li
i=1 i=1 i=1
nn n n
+6ΣU3Xi'(∑XiXiy ΣU3Xi + 24∑(Ui — 3)2∙
i=1 i=1 i=1 i=1
19
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