Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 169

This result should not surprise us since the balance on the auxiliary variables
exactly corresponds to the calibration of these variables in the design phase and
not in the estimation phase, as it is instead done by the GREG estimator (see
Sect. 10.2 ).
2. It protects against large sampling errors, because the most unfavorable samples
have a null probability of being selected.
3. It protects against a misspecification of the model within a model-based infer-
ence (Royall and Herson 1973 ).
4. It can ensure that the sample sizes in planned domains are not too small, or even
equal to zero. By adding the indicator variables of the planned domains to the list
of balanced auxiliaries, we can fix the sample size for each domain.

However, it is in a spatial context that this sampling plan could be successfully
applied to fix the sample size for each domain when we need to exploit several
auxiliary information sources (regardless of its applications to business or agricul-
tural households). We can impose that, for any selected sample, the HT estimates of
the first M moments of each coordinate should match the first M moments of the
population. In this way, we implicitly assume that the survey variable y follows a
polynomial spatial trend of order M .
Using a similar consideration, Breidt and Chauvet ( 2012 ) extended the approach
using linear mixed models at the design stage to incorporate the available auxiliary
information. Modifying the variables to be included in the constraint in Eq. ( 7.6 ),
they suggested a variant of the cube method that draws penalized balanced samples .
Instead of directly using the covariates X, they generated a new ordered set of
variables B 1 using penalized splines, which can model linear and nonlinear trends
more efficiently than a simple polynomial approximation (Hastie and Tibshirani
1990 , p. 52). The subjectivity lies in choosing the order of the splines and number of
knots K .
Suppose that y follows a linear mixed model of the form

y ¼ X ʲ þ Z ʳ þ ʵ; ð 7 : 11 Þ

where

¼ 0, and Var ¼ ˃

ʻ 2 Q 0
0

ʳ
ʵ

ʳ
ʵ

2

E ; ð 7 : 12 Þ

I

X is a full rank Nq matrix, Z is a full rank NK matrix, and I denotes an
appropriately sized identity matrix. We suppose that Q is a positive definite and
known matrix, typically, but not necessarily, an identity matrix. The parameter ˃

2 is

unknown and the parameter can be interpreted as a penalty for model complexity,

which the user must fix in advance. If

ʻ

2

¼0, there is no penalty for model
complexity, and Eq. ( 7.11 ) is a simple regression on X and Z. Conversely, if
ʻ

ʻ

2 2

! 1 the model is a regression on

> 0 the model will smooth y. Finally, if ʻ

only X (Breidt and Chauvet 2012 ).