Agriculture Reference
In-Depth Information
If the
N
dh
are known for
h
¼1,
...
,
H
, a count-synthetic estimator is a particular
t
case of Eq. (
11.19
) with x
d
¼
ð
x
d
1
...
x
dh
...
x
dH
Þ
;
and
x
dh
¼ 1if
d 2 U
h
or
x
dh
¼ 0 otherwise. It is defined as
sin
t
d
,
COU
¼
X
h
;
t
:h
N
dh
=N
:h
ð
11
:
21
Þ
dir
where
N
:h
is the estimator of the post-stratum size
N
.h
.
The following
R
code calculates the estimates using Eq. (
11.21
).
>
dtab
<
- table(framepop$coddom,framepop$q2obs)
>
dsize2
<
- cbind(as.numeric(rownames(dtab)),dtab)
>
colnames(dsize2)
<
- c("coddom", "1", "2", "3", "4", "5")
>
ws¼rep(N/n,n)
>
framesrs
<
- cbind(framesrs[,1:7],ws)
>
domsyn
<
- pssynt(y¼yobs, sweight¼ws, ps¼q2obs, domsizebyps¼dsize2,
+
data¼framesrs)
>
domsyn
Domain PsSynthetic
11
11
80.88565
12
12
92.84688
13
13
83.40660
21
21
88.45749
22
22
125.52893
23
23
95.61947
31
31
78.23234
32
32
89.15129
33
33
81.22648
These methods are currently used in practical applications because they do not
require a large computational effort, they do not require prior estimates at a small
domain level, and the independent variables are easily available from census,
satellite, or other administrative data. Unfortunately, the synthetic estimators are
biased, because they depend on strong assumptions. Hence, full MSE (accounting
for bias and variance) is relevant.
For example, the variance of
sin
t
d
is easily estimated (see Rao
2003
), but it is
more difficult to estimate the
MSE
of
sin
t
d
:
An approximately unbiased estimate of
was given by (Ghosh and Rao
1994
)
MSE
sin
t
d
¼
sin
t
d
dir
t
d
2
;
MSE
sin
t
d
V
dir
t
d
ð
11
:
22
Þ
is the design-unbiased estimator of the variance of the direct
V
dir
t
d
where
estimator
dir
t
d
:
Unfortunately, the estimator in Eq. (
11.22
) may be very unstable.
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