Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 115

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