Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 268

areas increases. However, from the theoretical point of view it is also possible that
spatial dependence increases as the size of the geographical unit decreases (see
Palma and Benedetti 1998 ). Conversely, the zone effect is the variation in numer-
ical results due to different zone systems, which can be obtained by merely
adjusting the boundaries at a certain analysis scale. In this case, the correlation
between variables is sensitive to the way that the analyst defines the different
regions.
From these considerations, it is evident that the SAs can be chosen according to
some administrative criteria or through economic relationships, bearing in mind
that different choices lead to different results and that the analyses should be
interpreted with particular caution.
In small area estimation (SAE) problems, the point estimates and error measures
are required for each area. A traditional approach for small domains is based on
classical design-based survey sampling methods (see Chap. 6 ). Estimates based on
this approach are often called direct estimates, and are generally obtained without
operationally using auxiliary data.
Unfortunately, direct area-specific estimates may not provide acceptable preci-
sion at the SA level. We expect them to have large standard errors because of the
small size (sometimes zero) of the sample in some areas. Furthermore, the direct
estimators cannot be calculated when there are no sample observations in some of
the relevant small domains. For these reasons, we must define appropriate pro-
cedures for estimating the characteristics of these SAs. The definition of these
ad-hoc alternative estimators is the objective of SAE techniques.
The use of SA statistics dates back to the eleventh century in England and the
seventeenth century in Canada (Brackstone 1987 ). In recent years, there has been a
lot of interest in SAE, which is justified by its increasing use for defining policies,
allocating funds, and regional planning.
SAE methods are often applied to statistical analysis for agriculture. As outlined
in the previous chapters of this topic, agricultural information is generally obtained
using sample surveys. However, in many national agricultural surveys estimates are
desired for small geographical areas. For example, operational SA information on
crop statistics is needed to formulate agricultural policies.
The issue of SAE is twofold. The first research question is concerned with
producing reliable estimates of characteristics of interest (i.e., totals, means, counts,
quantiles) for SAs or domains, based on very small samples taken from these areas.
The second question is how to evaluate the sampling error of these estimates.
Define a partition of the population U ¼ f 1 ; 2 ; ...; k ; ...; N g into D small

sub-domains U 1 , ... , U d , ... , U D ,with N d the size of U d . Then (S¨rndal et al. 1992 )

N ¼ X

D

D
d ¼1

U ¼ [ U d ; N d : ð 11 : 1 Þ

d ¼1

We assume that the parameter being estimated is the total at the small domain level
defined as