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distance. It can be seen in Eq. ( 1.29 ) that we can increase the efficiency of the HT
estimator by defining a design in which the
π kl are higher for any pair k,l such that
y k /
π l are very different. The formal expressions of Eqs. ( 1.29 ) and ( 1.38 )
are very similar; the main difference lies in the factor
π k and y l /
Δ kl that weights any pair k,l
and can be modified according to the sample design.
Other motivations for using the spatial nature of the population can arise by
introducing a model that links y with the auxiliaries X using some form of spatial
dependence (see Sect. 1.4.3.2 ). In this case, the anticipated variance of the HT
estimator of the total of a variable y given X is defined in Eq. ( 8.32 ) of Sect. 8.5 .Itis
clearly minimized when the dependence parameter ( ˁ kl ) between each pair k,l is
equal to 0. Assuming that
ˁ kl decreases as the distance between k,l increases, to
minimize the sampling variance, we should select the sampling units so that we
maximize the distance between them. In other words, the sample should be well-
spread over the study region.
There are other practical reasons for spatially well-distributed samples. First, it
appears to be appropriate when the mean and/or variance of y has zones of local
stationarity. In other words, if there is a spatial stratification in the observed
phenomenon. Finally, a well-spread sample is convenient when the coordinates of
the population can be expressed using a spatial point pattern that is clustered (i.e.,
the intensity of the units varies across the study region) (see Sect. 1.4.3.2 ).
7.3 Sampling Plans that Exclude Adjacent Units
In the literature regarding survey methodology, one of the topics of foremost
interest is how to improve estimations of population characteristics using some
additional knowledge of the sampling units. This efficiency gain is generally even
more noticeable when the enhancements are applied to the sample design, rather
than the estimator. If there exists some ordering of the units, and contiguous units
are anticipated to provide similar data, Hedayat et al. ( 1988b ) suggested that more
information could be obtained if the sample avoids pairs of contiguous units.
It is interesting to note that this feature is considered so important that it was
suggested by Hedayat et al. ( 1988b ) as a practical solution. In fact, they observe that
... if in any observed sample contiguous (or close to each other in some sense) units occur,
they may be collapsed into a single unit with the corresponding response as the average
observed response over these units. An estimate of the unknown parameter is then made on
the basis of such a reduced sample”.
Introducing the methodological aspects of this issue, they recognized that it is
advisable to use a sampling design with second-order probabilities that do not
decrease according to the distance between units. This possibility is extremely
limited by the practical evidence that there is no selection algorithm that respects
these required
π kl s.
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