Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 286

g 1 d

þ g 2 d þ g 3 d

MSE ʸ d , EBLUP

2
˅

2
˅

2
˅

˃ ˃ ˃ ð 11 : 39 Þ

where:

¼ ʳ d ˈ d < ˈ d ;

2
˅

g 1 d ˃

" # 1

2 x d X
d

¼ 1 ʳ d

2
˅

x d x d = ˈ d þ ˃

2
˅

b d g 2 d ˃ ð Þ x d ,

¼ ˈ 3 V ;

2
˅

2

d b d = ˈ i þ ˃

2
˅

b d

2
˅

g 3 d ˃ ˃

is the asymptotic variance of an estimator of

and V

˃

2
˅

˃

2
˅

.

may be

) shows that the MSE ʸ d , EBLUP

2
˅

Note that the main term g 1 d ( ˃

considerably smaller than MSE ʸ d if the weight

2
˅

ʳ d is small, or if ˃ is small

compared with ˈ d . This means that the SAE process largely depends on the

availability of good auxiliary information for reducing the model variance

2
˅

˃

with respect to ˈ d , as we would expect. Finally, Rao ( 1999 ) considered the

jackknife estimate of the MSE as a possible alternative.
The EBLUP estimates can be calculated in R using the sae package. We assume
that an auxiliary variable is available, as generated by the following code.

> set.seed(160964)
> auxFH < - 0.5*tapply(framepop$yobs,framepop$coddom,mean) +2*rnorm
+ (nrow(dsize))
> datFH < - cbind(domdir,dsize,auxFH)
> datFH$var < - datFH$se^2
The following instruction calculates the EBLUP estimates.

> domFH < - eblupFH(yobs ~ auxFH - 1, vardir ¼ var, data ¼ datFH, method ¼
+ "REML")

> domFH
$eblup

[,1]
11 77.28485
12 96.07419
13 74.66059
21 96.31286
22 119.19193
23 96.10808
31 78.57662
32 95.09622
33 82.04489
$fit