Agriculture Reference
In-Depth Information
obtained (Berger and Till
´
2009
). A rejective procedure consists of a conditioning
Poisson sampling design with respect to a fixed
n
.
Poisson sampling (PS) is a very simple
ps
procedure without replacement. Its
main drawback is that the actual sample size may be very different from the desired
n
. Ohlsson (
1998
) proposed sequential Poisson sampling (SPS) as a modification to
the original PS procedure, which corrects for this problem.
A random number uniformly distributed between 0 and 1,
R
k
~
U
[0,1], is asso-
ciated with each population unit
k
. Then, we can transform these random numbers
using
ˈ
k
¼ (
R
k
/
π
k
)
n
. A sample is obtained from the
n
units corresponding to the
n
smallest
ˈ
k
.
This very simple selection criterion maintains a fixed sample size, but unfortu-
nately SPS is not strict
π
ps
. However, simulation results obtained by Ohlsson
(
1998
) suggest that SPS is approximately
π
ps
. This means that it can be used if
we do not have very accurate procedures, or when simplicity is important. Its
variance can be estimated using
π
2
X
k2s
1
1
nn
y
k
V
SPS
t
SP
ðÞ
¼
p
k
t
SPS
ð
np
k
Þ
;
ð
6
:
9
Þ
ð
1
Þ
!
.
X
π
k
¼
π
k
/
n,
and
t
SPS
¼
where
ðÞ
1
=
n
ð
y
k
=
p
k
Þ
k2s
In the
sampling
R
Package, several algorithms have been implemented to
select a sample with a
ps
. We have used the
UPtille
function, which uses the
split pivotal method (Deville and Till
´
1998
). It is not the quickest method, but is
probably the most accurate. The selected sample is reported in Fig.
6.3
.
π
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