Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 269

t d ¼ X U d y k , d ¼ 1, ... , D ; ð 11 : 2 Þ

or the domain means

y d ¼ t d = N d , d ¼ 1, ... , D : ð 11 : 3 Þ

Note that if N d is unknown, y d is a ratio of two unknowns. The general notation for
the estimators of these parameters are t d and y d , respectively.
It is worth noting that the division in SAs can be viewed as a new method for
partitioning the population into subsets. This can be compared with the concepts of
strata and clusters in two-stage sampling, as discussed in Chap. 6 .
There have been many contributions to SAE research. In particular, an interested
reader can refer to Ghosh and Rao ( 1994 ), Rao ( 1999 , 2002 , 2003 ), and
Pfeffermann ( 2002 , 2013 ). These monographic papers highlighted the main theo-
ries and/or methodologies for practical SAE.
It is worth noting that SAE is very different from areal interpolation in terms of
data availability and methodologies, although both approaches deal with
spatial data.
Areal interpolation is the process of estimating the values of one or more
variables in a set of target areas, based on known values that exist in a set of source
areas. We need areal interpolation when data from different sources are collected in
different areal units. In the United States, for example, spatial data that have been
collected in census zones and tracts are very common. If the researcher wishes to
analyze marketing zones, the census data can be aggregated using a certain method
that takes advantage of natural or commercial (rather than administrative) bound-
aries. Areal interpolation techniques are necessary to transform the original data
into the so-called transformed data. There are many different methods of areal
interpolation. Each method is unique in its assumptions about the underlying
distribution of the data. Areal interpolation methods treat the data as they are,
without considering if they are from a sample or census (see also Sect. 12.3 ). See
Tobler ( 1979 ), Goodchild and Lam ( 1980 ), Palma and Benedetti ( 1998 ), and
Hawley and Moellering ( 2005 ) for an outline of this topic.
Additionally, SAE methods consider the nature of the data. An essential part of
their methodology is to take advantage of sample data, and consider sampling errors
together with a specific model type area-random component.
The layout of this chapter is as follows. Section 11.2 describes the direct and
indirect estimation methods. Section 11.3 contains a review of the foremost SA
approaches, namely the area level and unit level models. Section 11.4 outlines
estimation techniques for the SA models in Sect. 11.3 . In Sect. 11.5 we describe the
spatial approach for SAE, and in Sect. 11.6 we discuss the issue of benchmarking.
Finally, the last section concludes the chapter. The main R codes for the methods
presented in the chapter are also provided, with applications to both artificial data
and agricultural surveys.