Estimation of Superpopulation Parameters
191
( )
( )
∂
∂
=
∂
∂
Cov E
log
, , E
log
,
,
n
r
s
f A
f A
x
u
u
⌳
⌳
( )
( )
∂
∂
∂
∂
+E
log
, E
log
,
n
n
r
s
f A
f A
x
x
u
u
(A.49)
Combining Equations A.47, A.48, and A.49, we have
( )
2
2
1
log
E
log
,
N
k
n
k
s
r
s
r
L
fY
=
∂
∂
=
∂ ∂
∂ ∂
∑
x
u u
u u
(
)
( )
(
)
Cov
log
,
log
,
,
n
n
r
s
N n
f A
f A
∂
∂
+ −
∂
∂
x
x
u
u
(
)(
)
( )
1 Cov E
log
, ,
r
N n N n
f A ⌳
∂
+ −
− −
∂
u
( )
E
log
,
,
n
s
f A ⌳
∂
∂
x
u
(A.50)
But the last covariance term is equal to
(
)
(
)
1
2
Cov
log
,
log
,
n
n
n
r
s
f Y
f Y
+
+
∂
∂
∂
∂
x
u
u
where the conditional joint density of (Y n11 , Y n12 ) given x n is obtained
from Equation A.23 as
(
)
(
) (
) (
)
«
∞
= ∫
1
2
1
2
0
,
,
,
,
,
n
n
f a a
a
a
d
x
x
h
l h
l
l
l
(A.51)
Therefore, the (r, s) entry of I 0 () given by Equation A.21 is
( )
( )
( )
( )
2
0,
1
1
1
E
l o g
,
Cov
log
,
log
,
N
rs
k
n
k
s
r
N
N
k
j
n
k
j
r
s
f Y
f Y
f Y
=
=
=
−∂
=
∂ ∂
∂
∂
−
∂
∂
∑
∑
∑
I
x
x
u u
u
u
(A.52)
The observed Fisher information matrix at is the difference of the
conditional expectation of the complete-data information matrix and
the conditional covariance of the complete-data score functions, given
the data x n . In the case of the regular exponential family in Equation
A.45, the observed Fisher information matrix is
( )
( )
=
−
0
Cov ( )
Cov
,
n
N
N
n
t
t
I
x
Y
Y x
(A.53)
191
( )
( )
∂
∂
=
∂
∂
Cov E
log
, , E
log
,
,
n
r
s
f A
f A
x
u
u
⌳
⌳
( )
( )
∂
∂
∂
∂
+E
log
, E
log
,
n
n
r
s
f A
f A
x
x
u
u
(A.49)
Combining Equations A.47, A.48, and A.49, we have
( )
2
2
1
log
E
log
,
N
k
n
k
s
r
s
r
L
fY
=
∂
∂
=
∂ ∂
∂ ∂
∑
x
u u
u u
(
)
( )
(
)
Cov
log
,
log
,
,
n
n
r
s
N n
f A
f A
∂
∂
+ −
∂
∂
x
x
u
u
(
)(
)
( )
1 Cov E
log
, ,
r
N n N n
f A ⌳
∂
+ −
− −
∂
u
( )
E
log
,
,
n
s
f A ⌳
∂
∂
x
u
(A.50)
But the last covariance term is equal to
(
)
(
)
1
2
Cov
log
,
log
,
n
n
n
r
s
f Y
f Y
+
+
∂
∂
∂
∂
x
u
u
where the conditional joint density of (Y n11 , Y n12 ) given x n is obtained
from Equation A.23 as
(
)
(
) (
) (
)
«
∞
= ∫
1
2
1
2
0
,
,
,
,
,
n
n
f a a
a
a
d
x
x
h
l h
l
l
l
(A.51)
Therefore, the (r, s) entry of I 0 () given by Equation A.21 is
( )
( )
( )
( )
2
0,
1
1
1
E
l o g
,
Cov
log
,
log
,
N
rs
k
n
k
s
r
N
N
k
j
n
k
j
r
s
f Y
f Y
f Y
=
=
=
−∂
=
∂ ∂
∂
∂
−
∂
∂
∑
∑
∑
I
x
x
u u
u
u
(A.52)
The observed Fisher information matrix at is the difference of the
conditional expectation of the complete-data information matrix and
the conditional covariance of the complete-data score functions, given
the data x n . In the case of the regular exponential family in Equation
A.45, the observed Fisher information matrix is
( )
( )
=
−
0
Cov ( )
Cov
,
n
N
N
n
t
t
I
x
Y
Y x
(A.53)