Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 284

logit model appears to be appropriate. However, qualitative variables are very
uncommon in a sampling design for polygons or areas.

11.4 Estimation for Small Area Models

Various approached have been proposed for estimating the SAMs described in
Sect. 11.3 . The most commonly used methods for model-based SAE are: empirical
Bayes (EB), hierarchical Bayes (HB), and empirical best linear unbiased prediction
(EBLUP). The maximum likelihood (ML) and the restricted maximum likelihood
(REML) 1

techniques can be used to estimate variance components, assuming

normality.
The EBLUP method is appropriate for linear mixed models that include the basic
area and unit level models. Conversely, EB and HB techniques are more general,
and are applicable to generalized linear mixed models for categorical and count
data. In the case of normal linear mixed models, EB and EBLUP are identical, and
almost equal to the HB estimators. The EBLUP and EB approaches use MSE as a
measure of variability, while the HB approach uses the posterior variance assuming
a prior distribution on the model parameters.
In this section, we provide some technical details for the EBLUP estimator. For
more information regarding EB, HB, and EBLUP, see Rao ( 2003 ), Ghosh and Rao
( 1994 ), and Pfeffermann ( 2002 , 2013 ).
The EBLUP point estimator is not based on a distributional assumption. Con-
versely, the MSE estimation process assumes that the random effects ˅ i and e i are

Gaussian. Note that the estimator is described here with reference to the basic area
level model in Eq. ( 11.28 ).
The best linear unbiased prediction (BLUP) approach is often used to predict the
random or mixed effects of a SA model. The BLUP method was introduced by
Henderson ( 1950 ). A comprehensive overview of the derivations of the BLUP
estimator, with useful applications, is provided in Robinson ( 1991 ) and Rao
( 2003 ). The BLUP estimator for ʸ d under Model ( 11.28 ) is (Ghosh and Rao 1994 )

ʸ d , BLUP ¼ x d β þ ʳ d ʸ d x d β

¼ ʳ d ʸ d þ 1 ʳ d Þ x d β;

ð ð 11 : 35 Þ

b d
ˈ d þ ˃

˃

2
˅

where ʳ d ¼ , 0 ʳ d 1 is the shrinkage factor, ˈ d are the samples

b d

2
˅

is the weighted least square estimator of

variances, and β ˃

2
˅

β defined as

1 REML is a method (Patterson and Thompson 1971 ) in which estimators of parameters are
derived by maximizing the restricted likelihood (RL) rather than the likelihood itself. The RL is
calculated from a transformed data set, so that nuisance parameters have no effect.