Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 275

the survey package provides a more accurate variance estimation when the
sampling design is known. Furthermore, if the sampling design is not simple
random or Poisson sampling, sae calculates the estimated variances using the
approximation that second-order inclusion probabilities are the product of first-
order inclusion probabilities. Finally, the survey package is more flexible than
sae, because it allows for a more sophisticated sampling design (i.e., stratified,
two, or multi-stage sampling). For these theoretical and practical reasons, we
suggest that the survey package is used for direct estimates in SAs (Lumley
2010 ).
Now, assume that we have information about the auxiliary variables in the form
of known population totals for each SA d and that for each unit k in the sample s ,
x k ¼

t is observed. Then, the GREG estimator (see Sect. 10.2
for details) is defined as (S¨rndal et al. 1992 )
t d , GREG ¼ X U d ^

x k 1 x k 2 ... x kq

ð Þ

y k þ X s d ½ ð y k ^ y k Þ=π k ; ð 11 : 13 Þ

1

X s d X s d

x k x t k
˃

x k y k
˃

y k ¼ x k B d ,

B d ¼

2

where ^ ; and V ʾ yðÞ ¼ ˃

d :

The

2 2

d π k d π k

estimator in Eq. ( 11.13 ) is appropriate if N d is unknown. Otherwise, we can use the
alternative estimator
t 0 d , GREG ¼ X U d y k þ N d =N d

X s d

ð y k y k Þ=π k ; ð 11 :

14 Þ

½

where N d ¼ X s d Note that the GREG estimator can be negative for some

SAs, if the linear regression overestimates the variable of interest. The GREG
estimator is approximately design-unbiased for SAE, but it is not consistent because
it has large residuals. See Rao ( 2003 ) and S¨rndal et al. ( 1992 ) for a comprehensive
discussion on the use of the GREG estimator on small domains.
An additional way of improving the accuracy of estimates is to again consider
the auxiliary variables X, and to use SAE indirect methods based on implicit
models. These traditional indirect methods are generally considered design-based,
and their variances are usually smaller than those of the direct estimators
(Pfeffermann 2013 ).
To define this group of estimators, we must subdivide the D SAs according to
H groups or strata defined through X. Additionally, the sample s is analogously
divided into small sub-domains. In this way, the population and the sample are
partitioned into a grid of D H sub-populations and subsamples. Then, Eqs. ( 11.1 )
and ( 11.5 ) can be re-written as

ð 1 =π k Þ: