Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 113

π ps designs are considered. A good

introduction to pps can be found in S¨rndal et al. ( 1992 , pp. 97-100).
Let us consider the HT estimator (see Eq. (1.22) of Sect. 1.2 ) of the unknown
population total t . If we can implement a without replacement selection procedure
for fixed n such that y k /

ps and pps . For the purpose of this topic, only π

π k ¼ c , where c is a constant, then for every sample s we will

have that t HT ¼ nc . Since t HT is constant and does not depend on s , its variance will

be zero. This occurrence is obviously only theoretical. There is no design that
exactly satisfies this proportionality requirement, because it implies the full knowl-
edge of y k . However, if we can assume that we know a variable x that is approx-
imately proportional to y, then

π k can be chosen proportional to the known values

x k . Now, the ratios y k / π k will be as close to constant as the accuracy of the

proportionality estimate and, as a consequence, the variance of t HT will be small.
Let ( ʻ 1 , ... , ʻ k , ... , ʻ N ) be the objective set that represents the inclusion proba-
bilities of the selection rule that satisfies 0 k and X U ʻ k ¼

< ʻ k 1, 8 n . Thus, to

select a ps sample, we must determine a selection algorithm where the inclusion

probabilities are approximately, or asymptotically,

π

π k ʻ k , 8 k . The first problem

concerns the size of x. In fact, even if x k > 0, 8 k , it may not be trivial to determine a

k , particularly if we are dealing with highly
skewed populations. The required proportionality factor can be defined as

π k / x k , 8

set of inclusion probabilities

n X U x k , and so nx k X U x k .

Obviously, our requirement is that

π k ¼

π k 1. If n ¼ 1, the condition is satisfied 8 k .

nx k X U x k > However, if n > 1, some values of x k could result in π k ¼ 1, and

therefore 1. This problem can be solved using a census stratum ( A ) that

contains the largest units of the population (Benedetti and Piersimoni 2012 ).
More formally, we can define

π k >

X U x k

π k ¼ 1if k : nx k >

; ð 6 : 8 Þ

π k / x k otherwise

where the proportionality in the sampled stratum should be evaluated using
π k ¼

X UA x k

x k

ð n n A Þ for k 2 U A , to retain n . This rule is often not

sufficient to solve the problem, because the new proportionality coefficient
may generate new units k such that 1. Thus, the rule in Eq. ( 6.8 ) should

be iteratively applied, because every unit will have an inclusion probability
lower than 1.

π k >

> ypps < - exp(yobs/10)
> cor(yobs,ypps)
[1] 0.6915849
> summary((ypps/sum(ypps))*n)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000276 0.0046890 0.0243500 0.1000000 0.1212000 1.1890000