Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 262

Fig. 10.3 One population included within another ( left ), and two populations with an overlapping
zone ( right )

although it has many benefits, by using land as a criterion for defining the statistical
unit, we have the obvious disadvantage of not allowing for the collection of
economic, social, and administrative information. The target population of surveys
on these topics necessarily consists of farms or other legal units that can rarely be
unequivocally traced on the basis of the land.
For notational simplicity, we will now summarize estimators for the overlapping
dual frame survey depicted in Fig. 10.3 , where the domain AB is non-empty or at
least coincides with one of the two frames. Given two populations A and B ,we
assume that each element belongs to at least one of the two lists (for example,
A farms) and that it is always

possible to determine if a unit can be classified into: ( a ) units belonging only to
the frame A ,( b ) units belonging only to the frame B ,or( ab ) units belonging to both
frames. It is worth noting that, in this case, the survey is performed using two
independent and different frames. Furthermore, it is possible to observe the same
target variable on each sample selected from the two different frames. This situation
is very different from the previously described case of indirect sampling.
Table 10.1 defines some quantities for the estimators.
The total of the survey variable y can be split into the sum of its parts in each
domain, that is

¼ spatial frame of points or polygons, and B ¼

t y þ t y þ t ab

y

t y þ t y þ ʱ t ab

y

t ab
y

t y ¼ ¼ þ ð 1 ʱ Þ ; ð 10 : 40 Þ

where (0,1). Thus, if the population size for the three domains cannot be known

before data collection, we can intuitively estimate the total of y using

ʱ2

t HT , y ¼ t HT , y þ t HT , y þ ʱt a HT , y , A þ

Þt a HT , y , B ; ð 1 ʱ ð 10 : 41 Þ

where t a HT , y , A and t a HT , y , B represent the estimators in domain ab calculated on the
basis of units that fall within ab and selected from A or B , respectively. It should be