Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 278

The synthetic estimator was first applied in 1968, when the National Center of
Health Statistics of the United States used the synthetic estimation for providing
state estimates of some health characteristics from the National Health Interview
Survey (NHIS). In agricultural surveys, Singh and Goel ( 2000 ) used a synthetic
approach to estimate crop yields in India at the Tehsil level, where the Tehsil is an
Indian subnational administrative unit.
However, as the SA sample size increases, a direct estimator becomes more
suitable than a synthetic estimator because it is design-unbiased. This consideration
leads to an alternative method that considers a weighted sum of the direct and
synthetic estimators: the weighted estimator is referred to as the composite estima-
tor. Using this approach combines the advantages and disadvantages of the direct
and the synthetic estimators. In fact, the composite estimator attempts to mediate
the potential effects of the bias of the synthetic estimator with the instability of the
direct estimator. The composite estimator can be defined as

t d ¼ ˉ d dir t d þ 1 ˉ d

Þ sin t d ;

ð ð 11 : 23 Þ

com

where dir t d is a direct estimator, sin t d

is a synthetic estimator, and ˉ d is a weight

between 0 and 1.
The main research question regards finding the optimal weights ˉ d . They can be

obtained by minimizing the MSE of the estimator in Eq. ( 11.23 ), assuming that
Cov sin t d ; dir t d

: 0, as follows

MSE sin t d

ð Þ

ˉ d ¼

: ð 11 : 24 Þ

ÞþV MSE sin t d

dir t d

ð

The optimal weight ( ˉ d ) can be computed using the estimator in Eq. ( 11.22 ) as the

2

sin t d dir t d

numerator and as the denominator (Rao 2003 ). However, this

ˉ d can be very unstable.
Another approach to composite estimation is to use a common weight

estimate for

ˉ d ¼ ˉ ,

and then minimize the MSE with respect to (Purcell and Kish 1979 ). This

estimator is often called the James-Stein estimator (JS, James and Stein 1961 ).
The JS approach has been generalized by Efron and Morris ( 1975 ). However, the
resulting composite estimator may be less efficient than the direct estimator for
some small domains (Rao and Shinozaki 1978 ).
For more details about these composite methods, see Purcell and Kish ( 1979 ),
S¨rndal and Hidiroglou ( 1989 ), and Rao ( 2003 ). Additionally, Eklund ( 1998 ) used a
composite approach to estimate net coverage error for the 1997 US Census of
Agriculture at the state (i.e., SA) level.
The sae package calculates a particular composite estimator (namely the
sample size dependent estimator) that was introduced by Drew et al. ( 1982 ). The
authors proposed an estimator that uses the following weight for the composite
formula in Eq. ( 11.23 )

ˉ