Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 270

11.2 Direct and Indirect Estimation Methods

Each SA problem must be carefully considered to ensure that the techniques used
address the particular problem under investigation. The choice of SA method
depends on the availability of auxiliary data and on the relationship between these
data and the variables of interest at the SA level. In practice, we are trying to borrow
information from the auxiliary data to increase the accuracy of the estimates.
SAE can be addressed using different estimation methods, which we can broadly
classify into two main groups:

1. SAE direct methods only use data from sample units belonging to the area of
interest, gathered during the current survey. These methods usually lead to
unacceptably large variances. Consequently, they are rarely used unless the
sample size is large enough for the small domain of interest.
2. SAE indirect methods use data on the variable of interest collected in related
small areas or at other time periods. They use auxiliary information such as
census data, administrative records, and/or satellite data.

Furthermore, SAE approaches can also be divided into design-based and model-
based methods. Design-based methods may use a model for the construction of the
estimators (referred to as a model-assisted approach). But the bias, variance, and
other properties of the estimators are considered under the design-based distribu-
tion, or in other words over all possible samples that could be selected from the
target population (i.e. randomization) with the population measurements consid-
ered as fixed values (see Sect. 10.3 ). Conversely, model-based methods usually
condition on the selected sample, and the inference is performed with respect to the
underlying model (see Sect. 10.2 ).
A common characteristic of several SAE approaches is the use of covariate
information. Some estimators only need to know the covariates of the sampled
units, and the true area means of these covariates. Other estimators require the
covariates for every unit in the population.
More formally, let s d denote the part of sample s that is contained in U d , that is

s d ¼ s \ U d ; ð 11 : 4 Þ

and let n s d be the size of s d . The corresponding sample equations of Eq. ( 11.1 ) are

n s ¼ X

D

D
d ¼1

s ¼ [ s d ; n s d : ð 11 : 5 Þ

d ¼1

The direct approach to SAE is design-based, and only depends on the data available
for the target area of interest. Although it is rarely used (and only if the samples are
sufficiently large to cover all the SAs with sufficient data in each area), different