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> probinc < - inclusionprobabilities(ypps,n)
> summary(probinc)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000277 0.0047060 0.0244400 0.1000000 0.1216000 1.0000000
> framepop < - cbind(framepop,probinc)
ps selection rule should satisfy:
1. The sample selection must be easily implemented.
2. The first-order inclusion probabilities must be proportional to x k .
3. The second-order inclusion probabilities must satisfy
π
It is often required that a
π kl >
0,
8
k
6 ¼
l .
Condition 3 implies a measurable sampling design according to S¨rndal
et al. ( 1992 ), and is necessary and sufficient for the existence of a variance
consistent estimator of the total.
A measurable sampling design satisfies the following conditions:
1.
π kl must be exactly computable, and the complexity of their computation must
be low.
2.
Δ kl <
l , to ensure that the variance estimators in Eqs. ( 1.27 ) and ( 1.30 )of
Sect. 1.2 are not negative.
0,
8
k
6 ¼
One of the most common
π
ps designs is the systematic
π
ps (Brewer 1963 ).
Unfortunately, however, this approach cannot guarantee that
l unlike
other spatial sampling algorithms (see Chap. 7 ). The problems arising from this
peculiarity are considered in Sect. 7.4 , and represent a specific topic of the general
problem of variance estimation. This is discussed in Sect. 10.5 and deeply inves-
tigated in Wolter ( 2007 ).
Hanif and Brewer ( 1980 ) and Brewer and Hanif ( 1983 ) listed more than
50 criteria for selecting a
π kl >
0
8
k
6 ¼
ps sample. These methods, with the exclusion of
systematic drawing, are generally quite complex for practical situations in which
n
π
2. Ros´n( 1997 ) introduced order-sampling designs , a class of designs that are
an important contribution to
>
π
ps sample selection methods. Some interesting
approximate procedures for
ps selection have been developed, see Sunter ( 1986 )
and Ohlsson ( 1998 ), among others. However, there has been a lot of effective
research in this field and several algorithms have recently been proposed (Deville
and Till´ 1998 ; Bondesson and Thorburn 2008 ; Bondesson and Grafstr¨m 2011 ). In
particular, Till´ ( 2006 , Chaps. 5 - 7 ) presented a comprehensive and updated refer-
ence on recent developments in this area.
For example, one method for unequal probability sampling is the maximum
entropy design. This plan is identical to the conditional Poisson design, but they
are obtained from two different perspectives (Berger and Till´ 2009 ). The
maximum entropy design is the design that maximizes the entropy of the sample
ls
π
X s ps
over all the samples of fixed size n that are subject to the
given inclusion probabilities
ðÞ ¼
½
ðÞ
log ps
ðÞ
π k . It can be simply implemented using a Poisson
rejective procedure, i.e., by reselecting Poisson samples until a fixed sample size is
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