5. Domain estimation
Takis Merkouris
Previous | Next
Composite
estimators for domains (subpopulations) of interest may be readily obtained
using the calibrated weights derived in the previous sections, that is, by
summing the weighted values of a variable over any domain
For instance,
letting
denote the
matrix
for sample
with the entries
of the
row set equal to
0 if
the CGR
estimator of the domain total
based on the
weights of
calibrated with
the scheme of design (c) (see Section 3) is given by
where
and the
subscript
indicates
domain. The CGR estimator
based on sample
is obtained in
the same manner. However, unlike the population-level estimator (3.2),
resulting from calibration of two estimators to each other at population level,
the estimators
and
are not
constructed as composites of two domain estimators, based on samples
and
and they are not
identical. Moreover, although both
and
incorporate
information on
from samples
and
their
construction (non-customized at domain level) may entail some loss of
efficiency.
A
simple modification of the calibration procedure that leads to efficient
composite estimation for all totals of interest involves the augmentation of
the design matrix with columns defined at each domain level for the relevant
variables. Thus, for design (c) estimation of the domain total
involves the
augmentation of the design matrix
in (2.7) with
the column
The resulting
estimator,
may be written
in the forms
where
and
are now the GR
domain estimators incorporating the regression effect of the second and third
terms of (5.1). Adding another term in (5.1) involving the difference
may not improve
appreciably the efficiency of
but will be
necessary if estimation of the domain total
is also
required. In any particular situation, the augmentation of the design matrix
involves only
those components of
or
for which domain
estimates are needed. A possible drawback of this procedure is the additional
computational burden, which increases with the number of domains and the
variables for which domain estimation is required.
An
alternative approach that may be more appropriate when the domain estimates of
interest are numerous, involves the separate production of the domain estimates
by carrying out the composite calibration only at the domain level. For the
domain total
this would give
the domain CGR estimator, in analogy with the population CGR estimator (3.2),
where
and
The efficiency
of the joint estimator
over the
estimator
can be verified
under the conditions of the following proposition (its proof in the Appendix).
Proposition 2 Under the sampling schemes of
Theorem 1,
Notably,
the drawback of a separate production of the domain estimates, through
composite calibration at the domain level, is the loss of consistency among
estimates at population level and domain level.
The
above considerations extend to domain estimation for matrix sampling design
(d).
Previous | Next