Agriculture Reference
In-Depth Information
The HT estimator of the total for a two-stage sample is
X
i2s
1
X
k2s
i
d
k
y
k
¼
X
i2s
1
t
1
i
,
HT
=π
1
i
t
HT
,
2
ST
¼
1
X
i2s
1
X
k2s
i
π
1
i
π
2
k
i
¼
y
k
;
ð
6
:
14
Þ
where
t
1
i
,
HT
is the HT estimator of
t
i
with respect to stage two,
π
2
kji
are the
first-order inclusion probabilities for the first-stage and the second-stage, respec-
tively. Following a similar logic with second-order probabilities, we obtain
(S¨rndal et al.
1992
, p. 137)
π
1
i
and
XX
i
,
j2s
1
Δ
X
i2s
1
V
i
^
1
ij
t
1
i
,
HT
π
1
i
t
1
j
,
HT
V
HT
t
HT
,
2
ST
Þ
V
PSU
þ V
SSU
¼
ð
π
1
j
þ
π
1
i
;
ð
6
:
15
Þ
XX
s
i
Δ
=π
1
ij
,
^
1
ij
^
kl
i
^
kji
^
lji
,
V
i
¼
where
Δ
¼
Δ
1
ij
=π
1
ij
¼
π
1
ij
π
1
i
π
1
j
j
¼
Δ
2
klji
=π
2
klji
¼
π
2
klji
π
2
kji
π
2
lji
=π
2
klji
.
The results in Eqs. (
6.14
) and (
6.15
) can be easily extended to
r-stage
sampling,
because of the independence of each selection performed at each stage (S¨rndal
et al.
1992
, p. 145).
We can specify the random criterion used for each stage and, as a consequence,
we can substitute the exact value for the selection probabilities in Eqs. (
6.14
) and
(
6.15
). For example, if we perform a simple random cluster sampling, Eq. (
6.14
)
reduces to
^
2
kl
^
kji
¼
y
k
=π
kji
, and
Δ
j
i
X
k2s
1
d
k
y
k
¼
n
1
X
k2s
1
t
y
k
;
N
1
t
HT
,
Clus
¼
N
1
t
s
1
¼
ð
6
:
16
Þ
where
t
s
1
1 are the
direct
sampling weights
obtained as a product of the expansion from the units to the
group (equal to 1 because it is all the units censused) and from the selected clusters
to the population
U
(equal to
N
1
/
n
1
). With regard to the variance of the HT
estimator, Eq. (
6.15
) reduces to
is the mean of the cluster totals, and
d
k
¼
(
N
1
/
n
1
)
S
t
,
s
1
;
1
f
1
1
n
1
1
N
1
V
HT
t
HT
,
Clus
N
1
S
t
,
s
1
¼
N
1
ð
Þ
¼
ð
6
:
17
Þ
n
1
2
X
k2s
1
t
y
k
t
s
1
where
S
t
,
s
1
¼
. Note that in Eqs. (
6.16
) and (
6.17
)
k
represents the
n
1
1
group instead of the unit used in the rest of the topic.
Search WWH ::
Custom Search