Agriculture Reference
In-Depth Information
According to the assumptions, every sample has the same probability of being
selected. That is
:
N
n
ps
ðÞ
¼
1
=
ð
6
:
1
Þ
The first-order inclusion probability for unit
k
is the sum of the sample probabilities
for all samples that contain
k
. For this design, it is constant and equal to the
sampling fraction f
, defined as
n
N
¼
π
k
¼
f
8
k
2
U
:
ð
6
:
2
Þ
Therefore,
X
U
π
k
¼
n
. The second-order inclusion probability is defined as
n
N
n
1
π
kl
¼
1
8
k
6
¼
l
2
U
:
ð
6
:
3
Þ
N
X
U
y
k
for a finite population
U
. According to
estimator of the total for an SRS selected sample is
Our aim is to estimate the total
t
¼
X
U
I
k
y
k
=π
k
X
U
I
k
^
k
¼
X
s
^
k
¼
X
s
d
k
y
k
t
HT
,
SRS
¼
ð
Þ
¼
X
s
y
k
;
¼
ð
N
=
n
Þ
ð
6
:
4
Þ
where
d
k
¼ 1
=π
k
¼
n
are the
direct sampling weights
or
expansion weights
that
only use the design information to expand the sample outcomes to the population
U
.
Proceeding in a similar way, we can replace Eq. (
6.3
) in the general HT variance
estimator (see Eq. (1.28) Sect.
1.2
) to obtain the variance estimator of Eq. (
6.4
)
N
=
S
y
;
f
N
2
1
N
2
1
1
N
V
HT
t
HT
,
SRS
S
y
¼
ð
Þ
¼
n
ð
6
:
5
Þ
n
X
k2s
2
ð
y
k
y
Þ
where
S
y
¼
1
.
Denoting
p
(
.
) to be some other non-SRS design with the same sample size
n
,we
obtain the variance ratio
n
V t
HT
,
p
;t
HT
deff p
ð
Þ
¼
Þ
;
ð
6
:
6
Þ
V
HT
t
HT
,
SRS
ð
which is defined as the
design effect
. Note that the same approach can be used to
compare two different estimators. This index expresses how well the design and
estimator performs in comparison with the basic SRS strategy. When it is greater
Search WWH ::
Custom Search