Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 276

N ¼ X

D

X

H

D
d ¼1 [

H

U ¼ [ U dh ; N dh ; ð 11 : 15 Þ

h ¼1

d ¼1 h ¼1

n s ¼ X

D

X

H

D
d ¼1 [

H

s ¼ [ s dh ; n s dh : ð 11 : 16 Þ

h ¼1

d ¼1 h ¼1

The sizes n s dh are supposed to be random. Now, suppose that for each strata h, we
can use the survey data to calculate the direct estimator dir t :h of the strata totals of the
characteristic of interest t . h , where

t :h ¼ X d t dh ; ð 11 : 17 Þ

and t dh is the total of the variable of interest y for the units in the SA d that belong to
stratum h . We want to estimate the SA totals for d ¼1, 2,

... , D , defined as

t d ¼ X h t dh ; ð 11 : 18 Þ

using the known auxiliary total x dh .
The synthetic estimator is a popular indirect estimation approach. Gonzalez
( 1973 ) presented an appropriate definition for the synthetic estimator
“An unbiased estimate is obtained from a sample survey for a large area; when this estimate
is used to derive estimates for subareas under the assumption that small area have the same
characteristics as the large area, we identify these estimates as synthetic estimates”.

Additionally, Levy ( 1979 ) and Rao ( 2003 ) provided extensive synopses on
various synthetic estimation approaches, and their applications to SA estimation.
The regression-synthetic estimator is commonly used if domain-specific infor-
mation is available in the form of known totals x d. . It is defined for the d -th SA as

sin t d , REG ¼ x d B

; ð 11 : 19 Þ

where B is the sample weighted least squares estimator.
If there is a single auxiliary variable x , the synthetic estimator is defined as
(Ghosh and Rao 1994 ; Rao 2003 )
sin t d , RAT ¼ X h dir t :h x dh =^

ð x :h Þ; ð 11 : 20 Þ

where dir t :h is the reliable direct estimate of the strata total t .h , x dh is the known
auxiliary total for SA d that belongs to stratum h , and

x :h is the estimate of the
auxiliary total for stratum h . This estimator is also known as the ratio-synthetic
estimator, and represents a special case of the regression-synthetic estimator (Rao
2003 ).

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