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 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
1 ij ,
^ 1 ij
^ kl
i ^ kji ^ lji ,
V i ¼
where
Δ
¼ Δ 1 ij 1 ij ¼ π 1 ij π 1 i π 1 j
j
¼ Δ 2 klji 2 klji ¼ π 2 klji π 2 kji π 2 lji 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 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.
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