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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