Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 285

! 1 !

X X

¼

β ¼ β ˃

x d ʸ d = ˈ d þ ˃

2
˅

x d x d = ˈ d þ ˃

2
˅

b d

2
˅

b d : ð 11 : 36 Þ

d d

The estimator in Eq. ( 11.35 ) is the best linear unbiased estimator (BLUE). From
Eq. ( 11.35 ), it is evident that the BLUP estimator is a linear combination of the
direct estimator ʸ d ¼ ʸ tðÞ and the regression-synthetic estimator x d β:
ʸ d gives more weight to the synthetic estimator x d β

b d is
small. Conversely, more weight is given to the direct estimator ʸ d if the variance

2
˅

if the model variance ˃

ˈ d

is design-consistent. The BLUP estimator depends on

is small. Furthermore, ʸ d

2
˅

˃

2
˅

the variance component that is generally unknown in practical applications.

However, there are various techniques for estimating the variance components in
a linear mixed model: the method of moments (which does not require the normal-
ity assumption), the ML and the REML (which both require the assumption of
normality to estimate

˃

2
˅

). Jiang ( 1996 ) showed that the REML estimator remains
consistent when the normality assumption is relaxed. For a discussion about the
estimate of variance components see Cressie ( 1992 ).
Replacing

˃

2
˅

2
˅

, we obtain an empirical BLUP estimator known as the
EBLUP estimator (Harville 1991 ). It is unbiased if the distributions of

˃ with ˃

and e are
symmetric, but not necessarily normal. Note that the MSE of the EBLUP estimator
is essentially insensitive to the choice of

˅

2
˅

˃ . The EBLUP estimator is

ʸ d , EBLUP ¼ ʳ d ʸ d þ 1 ʳ d

Þ x d β;

ð ð 11 : 37 Þ

b d
ˈ d þ ˃

2
˅

˃

is the estimator of the shrinkage factor.

where ʳ d ¼

b d

2
˅

An important aspect of SAE is the assessment of the prediction errors. Assessing
prediction errors using the EBLUP approach is complicated, because of the errors
induced by the estimation of the model parameters. A variability measure associ-
ated with the EBLUP estimator is

2

MSE ʸ d , EBLUP ¼ E ʸ d , EBLUP ʸ d

: ð 11 : 38 Þ

Unfortunately, closed forms of MSE ʸ d , EBLUP only exist in particular cases. With
this in mind, many scholars have tried to identify accurate approximations for
Eq. ( 11.38 ).
If D is large and we can assume that the errors ˅ and e are normal, a valid

is (Rao 2003 , p. 128)

approximation for MSE ʸ d , EBLUP