Agriculture Reference
In-Depth Information
X
s
ˀ
k
u
k
. The maximum pseudolikelihood is the value
ʸ
PL
that represents the solution of the equation
sc
w
estimate is
sc
w
ðÞ¼
X
s
ˀ
1
ʸ
PL
¼
^
u
k
¼
0, where
k
.
One possible alternative likelihood-based approach for sample survey analysis
was suggested by Krieger and Pfeffermann (
1992
,
1997
) and Pfeffermann
et al. (
1998
). This method is the sample likelihood technique. This is a model-
based approach to analysis, which is different from the pseudolikelihood approach.
The sample likelihood approach is based on the estimates of the conditional
density (
f
(y
U
|X
U
)) parameters. The basic idea is that the population units are
considered independently conditional on X
U
. Using this assumption, we can write
log f y
k
; ʸ
PL
u
k
¼
∂
ʸ
^
Y
k2U
f
y
k
X
U
¼
;
f
y
U
X
U
j
j
ð
12
:
25
Þ
where
f
(y
k
|X
U
) denotes
the
conditional population density of
the
k
-th
population unit.
However, if we are using an informative sample selection method, we need to
introduce a conditional sample density for the
k
-th population unit. This can be
expressed as
fy
ks
X
U
ð
j
Þ¼
fy
k
I
k
¼
ð
j
1, X
U
Þ;
ð
12
:
26
Þ
where
y
ks
denotes the value of the target variable
y
that corresponds to a selected
unit
k
. Applying Bayes
Theorem, we obtain
'
Pr I
k
¼
ð
1
y
k
;
j
X
U
Þ
fy
k
X
U
ð
j
Þ
fy
ks
X
U
ð
j
Þ¼
:
ð
12
:
27
Þ
Pr I
k
¼
ð
1 X
U
j
Þ
As argued by Pfeffermann et al. (
1998
), when
N
is large and
n
is small relative to
N
,
it is realistic to assume that the sample units are also independently distributed
conditional on X
U
. Then, it is possible to write
Y
k2s
fy
ks
X
U
Y
k2s
¼
Pr I
k
¼
ð
1
y
k
;
j
Þ
fy
k
X
U
ð
j
Þ
X
U
f
y
s
X
U
j
ð
j
Þ¼
:
ð
12
:
28
Þ
Pr I
k
¼
ð
1 X
U
j
Þ
The density in Eq. (
12.28
) defines the sample likelihood for the parameters of the
conditional distribution of y
s
|X
U
.
If we are using a non-informative sample selection method, given X
U
, (i.e., if
¼
f
i
U
y
U
;
X
U
f
i
U
X
U
ð
j
Þ
), then the sample likelihood is
Y
k2s
fy
k
X
U
¼
f
y
s
X
U
j
ð
j
Þ:
ð
12
:
29
Þ
For a formal expression of the maximum sample likelihood estimate equations, an
interested reader can see Chambers et al. (
2012
, p. 65).
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