Spatial Survey Data Modeling - Sampling Spatial Units for Agricultural Surveys - page 306

We can derive the unit weights from the optimal estimator in Eq. ( 12.8 )to
calculate the linear estimator

ʸ ¼

g s t Y s . For a single target variable y, the unit

weights are

1 s þ

V 1
ss

X s A 1
s

X s X t s V 1 g s ¼ V ss

ss V ss

1 s ; ð 12 : 15 Þ

where 1 s and 1 s are, respectively, vectors of N n and n 1 s. Unit i is assigned a

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weight equal to the i -th component of vector g s .
Note that the estimator in Eq. ( 12.8 ) is asymptotically Gaussian under some
specific conditions (see Valliant et al. 2000 ).
In many empirical studies, some of the auxiliaries can be qualitative. Further-
more, quantitative variables can also be used in combination with qualitative
covariates. In the case of qualitative auxiliaries, under Model ( 12.2 ), the BLUP of
ʸ ¼ ʳ

t Y is

;

ʸ opt ¼ ʳ

t
s Y s þ ʳ

t
s X s β

o

V ss V 1

o

þ

ss Y s

ð X s β Þ ð 12 : 16 Þ

o

GX t s V 1 where

ss Y s with G a generalized inverse of A s (Valliant et al. 2000 ,
Theorem 7.4.1). The matrix G is not unique, but the predictor in Eq. ( 12.16 )is
insensitive to the choice of G. The corresponding error variance is

β ¼

¼

Var ξ ʸ opt ʸ

GX s t ʳ s :

ð

s X s

V ss V 1 V ss V 1

s V ss

V ss V 1

¼ ʳ

ss X s ss X s

ʳ s þ ʳ

ss V ss

12 : 17 Þ

Until now, we have assumed that the target variable Y is quantitative. However, we
are aware that in many agricultural surveys the target variable can be qualitative.
For example, the land use and/or land cover of a certain territory. This last case
cannot be treated in an analogous way to the qualitative covariates

case. Therefore,
Eqs. ( 12.16 ) and ( 12.17 ) cannot be used when the study variable is qualitative.
Defining a BLUP for this situation represents an important line of research for the
near future.
In the previous equations, the inference methods are calculated with respect to
the model, and they do not consider the selection plans to be random. In other
words, we have assumed that the sample selection method is completely irrelevant
when making inferences. Following this approach, the selection methods are called
ignorable or non-informative (Chambers and Skinner 2003 ). Otherwise, if the unit
selection cannot be ignored when making inferences, the selection method is called
non-ignorable or informative.
However, we can only assume non-informative selection methods in some
particular cases. Valliant et al. ( 2000 ) contains a very interesting discussion about
this topic and possible solutions to this problem.

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