Estimation of Superpopulation Parameters
193
unknown inclusion probabilities and then estimate N by an approximate Horvitz–Thompson-type estimator. His method requires solving a pair of transcendental equations that are symmetrical in form.
Barouch et al. (1985) proposed an alternative pair that is asymmetrical
and is competitive with Gordon’s pair.
The fourth approach is to postulate that N also has a superpopulation probability function P(•|g) indexed by a vector of parameters ␥ and
independent of the variate Y’s, then derive an EM algorithm for both
and ␥. The probability function P(N|␥) may be interpreted as a model
describing a random mechanism with regard to how N is generated or
it may be considered as a prior distribution in an empirical Bayesian
context. Here, the observations consist of x n and N ≥ n. The completedata log likelihood is
(
)
( )
( )
=
…
=
+
∑
1
1
log
, , ,
,
log
log P
N
N
k
k
L y
y N
f y
N
␥
␥ (A.54)
Let (', ␥') denote the current estimate of (, ␥) and let d 5 { x n , N ≥ n }
denote the data. The M-step is to maximize over (, ␥) the following
conditional expectation:
(
)
(
)
{
}
(
)
{
}
1
1
,
', ' E log
,..., ,
,
, ', '
E E log
,..., ,
,
, , ', ' , ', '
N
N
n
Q
L Y
Y N
L Y
Y N
N
=
=
d
x
d
␥ ␥
␥
␥
␥
␥
␥
(A.55)
Now, for l 5 0, 1, 2, …,
(
)
( )
(
)
(
)
=
+
…
= +
=
+
+
+
= +
∑
1
1
1
E log
, , ,
,
,
', '
log
log P
E log
,
', '
N
n
n
j
j
n
n
L Y
Y N
N n l
f x
n l
l
f Y
N n l
x
x
␥
␥
␥
␥
(A.56)
Therefore,
(
)
( )
( )
(
)
(
)
{
}
=
+
=
+
−
∑
1
1
,
', '
log
E log P
, ', '
+ E
E log
,
', ' , ', '
n
j
j
n
n
Q
fx
N
N n
f Y
N
d
x
d
␥ ␥
␥
␥
␥
␥
(A.57)
The necessary conditions for (, ␥) to be a maximizer of Q(, ␥|', ␥'),
are
(
)
(
)
,
', '
E
log P
, ', ' 0
i
i
Q
N
g
g
∂
∂
=
=
∂
∂
d
␥ ␥
␥
␥
(A.58)
193
unknown inclusion probabilities and then estimate N by an approximate Horvitz–Thompson-type estimator. His method requires solving a pair of transcendental equations that are symmetrical in form.
Barouch et al. (1985) proposed an alternative pair that is asymmetrical
and is competitive with Gordon’s pair.
The fourth approach is to postulate that N also has a superpopulation probability function P(•|g) indexed by a vector of parameters ␥ and
independent of the variate Y’s, then derive an EM algorithm for both
and ␥. The probability function P(N|␥) may be interpreted as a model
describing a random mechanism with regard to how N is generated or
it may be considered as a prior distribution in an empirical Bayesian
context. Here, the observations consist of x n and N ≥ n. The completedata log likelihood is
(
)
( )
( )
=
…
=
+
∑
1
1
log
, , ,
,
log
log P
N
N
k
k
L y
y N
f y
N
␥
␥ (A.54)
Let (', ␥') denote the current estimate of (, ␥) and let d 5 { x n , N ≥ n }
denote the data. The M-step is to maximize over (, ␥) the following
conditional expectation:
(
)
(
)
{
}
(
)
{
}
1
1
,
', ' E log
,..., ,
,
, ', '
E E log
,..., ,
,
, , ', ' , ', '
N
N
n
Q
L Y
Y N
L Y
Y N
N
=
=
d
x
d
␥ ␥
␥
␥
␥
␥
␥
(A.55)
Now, for l 5 0, 1, 2, …,
(
)
( )
(
)
(
)
=
+
…
= +
=
+
+
+
= +
∑
1
1
1
E log
, , ,
,
,
', '
log
log P
E log
,
', '
N
n
n
j
j
n
n
L Y
Y N
N n l
f x
n l
l
f Y
N n l
x
x
␥
␥
␥
␥
(A.56)
Therefore,
(
)
( )
( )
(
)
(
)
{
}
=
+
=
+
−
∑
1
1
,
', '
log
E log P
, ', '
+ E
E log
,
', ' , ', '
n
j
j
n
n
Q
fx
N
N n
f Y
N
d
x
d
␥ ␥
␥
␥
␥
␥
(A.57)
The necessary conditions for (, ␥) to be a maximizer of Q(, ␥|', ␥'),
are
(
)
(
)
,
', '
E
log P
, ', ' 0
i
i
Q
N
g
g
∂
∂
=
=
∂
∂
d
␥ ␥
␥
␥
(A.58)