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Appendix B
(D2)
For generalized least squares estimation (GLS), , and are obtained by
taking f to be:
(D3)
For asymptotically distribution-free estimation (ADF), , and are obtained
by taking f to be:
(D4)
where the elements of are given by Browne (1984, Equations 3.1–3.4):
() () () ()
()
() () () ()
()
() () () ()
()
() () ()
()
() () () ()
()
() () ()
()
.logtrlog
gggggggggg
gggg
KL
gggg
KL
gggg
ML
p
fff
μμ
μμ
−Σ
′
−+−−Σ+Σ=
−Σ=Σ
−−
xxSS
S,x;S,xS,x;,S,x;,
11
C
GLS
F
GLS
() ()
()
() () ()
()
[]
2
2
1
tr
ggggg
GLS
f Σ−=Σ
−
SSS;
1
C
ADF
F
ADF
() ()
()
() ()
[]
() () ()
[]
∑
=
−
′
−=Σ
−
G
g
ggggggg
ADF
f
1
)()(;
1
γσγσ sUsS
U
g()
∑
=
=
g
N
r
g
ir
g
g
i
x
N
x
1
)()(
1
()(
)
)()(
1
)()()(
1
g
j
g
jr
N
r
g
i
g
ir
g
g
ij
xxxx
N
w
g
−−=
∑
=
()()()(
)
)()()()()()(
1
)()()(
,
1
g
l
g
lr
g
k
g
kr
g
j
g
jr
N
r
g
i
g
ir
g
g
klij
xxxxxxxx
N
w
g
−−−−=
∑
=