Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 213

x k (S¨rndal et al. 1992 , 12.2.6) is important when
finding a solution. It is a function of the auxiliary X (otherwise it will disappear
from the AV ), and was introduced by Isaki and Fuller ( 1982 ). Using Model ( 8.27 ),
the heteroscedastic variance satisfies (S¨rndal et al. 1992 , 12.2.13)

The heteroscedastic variance ˃

˃ Þ ¼ X

N

1
ˀ k 1

x k ;

AV t GREG t

ð ð 8 : 28 Þ

k ¼1

where t GREG is the generalized regression estimator of the population total y given
the auxiliaries X. The generalized regression estimator will be introduced in Sect.
10.2 .
If we assume that Var ʾ ʵðÞ ¼ ˃

k and Cov ʾ ʵ k ; ʵð Þ ¼ ˃ k ˃ l ˁ kl , under Model ( 8.27 )
the AV of the HT estimator of the total of a variable y given X is (Grafstr¨m and
Till´ 2013 )

" ! t # 2

Þ E s X
k2s

ˀ k X
k2U

þ X
k2U

X

x k

l2U ˃ k ˃ l ˁ kl ˀ kl ˀ k ˀ l

AV t HT t

ð x k ʲ : ð 8 : 29 Þ

ˀ k ˀ l

Note that the first part of Eq. ( 8.29 ) is the error that the HT estimator makes when
estimating the true and known totals of the covariate weighted by the regression
coefficients. The second term represents the typical expanded covariance of the
indicator random variables [see Eq. ( 1.27 ) ] used in the HT variance, but weighted
by the correlation of the model residuals.
This superpopulation model can help us to better understand the impact of space
when designing a sample of geo-coded units (see Sect. 7.2 ). From Eq. ( 8.29 ), it is
clear that the best design would be balanced on the set of auxiliaries in such a way
that the first term will be equal to 0, and spatially balanced with sampling units so
far apart that we can assume that

ˁ kl ¼ 0 for each pair k 6 ¼ l within the sample.
In practice, particularly in stratified sampling, the solutions proposed in the
literature are not explicitly finalized to minimize the AV . They simply substitute
the moments of the variable of interest in the results described in previous sections
with the anticipated moments ( AM )ofy given X (Rivest 2002 ; Baillargeon and
Rivest 2009 , 2011 ).
Several alternatives to the linear regression model have been proposed. The
log-scale relationship should reduce the effects of heteroscedastic errors and skew
populations. A zero-inflated linear model is useful when dealing with household
surveys, because a unit can go out of business between the collection of the
X variables and the date of the survey (Baillargeon and Rivest 2009 , 2011 ;
Benedetti and Piersimoni 2012 ).
It is important to underline that, although the AM approach is based on the
assumption of a superpopulation model

, it does not necessarily presume that the
survey estimates are not design-based or that any inference made on the sample
does not respect randomization principles.

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