Agriculture Reference
In-Depth Information
[3,]
0
1
1
0
1
[4,]
0
1
1
1
0
[5,]
1
0
0
1
1
[6,]
1
0
1
0
1
[7,]
1
0
1
1
0
[8,]
1
1
0
0
1
[9,]
1
1
0
1
0
[10,] 1
1
1
0
0
Note that we have fixed the seed so that the results are reproducible.
The inclusion probability that the unit
k
will be included in a sample is denoted
Þ
X
s
'
k
π
k
¼ Pr
k 2 S
ð
Þ
¼Pr
I
k
¼ 1
ð
pðÞ:
ð
1
:
7
Þ
The term
s
k
means that the sum is extended over those samples that contain
k
. The
probability in Eq. (
1.7
) is also called the first-order inclusion probability. It is also
possible to define the second-order inclusion probability for the units
k
and
l
,
'
Þ
¼
X
s
'
k&l
π
kl
¼ Pr
k&l 2 S
ð
Þ
¼ Pr
I
k
I
l
¼ 1
ð
pðÞ:
ð
1
:
8
Þ
The term
s
k&l
means that the sum is extended over those samples that contain
k
and
l
. A given design
p
(.) has
N
quantities (
'
π
1
,
π
2
,,
...
,
π
k
,
...
,
π
N
) that constitute
the
set
of first-order
inclusion
probabilities,
and
N
(
N
-1)/2
quantities
ð Þ
that are the second-order inclusion probabilities.
Higher order inclusion probabilities can be calculated, but they do not have an
essential role in the definition of estimators or in variance estimation. Conse-
quently, we will not analyze them in this topic.
Consider, for example, a population composed of four units, identified by the
labels
U
¼ {1,2,3,4}. The sample space of all possible samples without replacement
of size
n
¼3is
π
12
,
π
13
,,
...
,
π
kl
,
...
,
π
N
1
,
N
S
¼
f
ð
1
;
2
;
3
Þ;
ð
1
;
2
;
4
Þ;
ð
1
;
3
;
4
Þ;
ð
2
;
3
;
4
Þ
g:
If we define the sample design using
ps
1
¼ 1, 2, 3
ð
Þ
¼ 1
=
9,
ps
2
¼ 1, 2, 4
ð
Þ
¼ 4
=
9,
ps
3
¼ 1, 3, 4
ð
Þ
¼ 3
=
9,
ps
4
¼ 2, 3, 4
ð
Þ
¼ 1
=
9
;
then the first-order inclusion probability for element
k
¼1is
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