2.3 Indirect Small Area Estimation
2.3.2 Composite Estimators
Composite estimation can be considered as a compromise between synthetic and
direct estimation. Generally, a synthetic estimator gives better results than the
traditional direct estimator when the area level sample sizes are relatively small,
whereas direct estimators perform better than the synthetic estimators for large
sample sizes. A weighted average of these two estimators is an alternative to choosing
one over the other to balance between bias and variance. This type of estimator is
commonly known as a ‘composite estimator’.
Composite estimation is a natural way to balance the potential bias of a synthetic
estimator against the instability of a direct estimator by choosing an appropriate
weight (Ghosh and Rao, 1994). The composite estimator of population total Yd for
small area d can be defined as
b
Yd,COM =φdYbd,HT + (1−φd)Ybd,SY N (2.17)
synthetic estimators of Yd. φd is a suitably chosen weight that lies between 0 and 1
(Rao, 2003b, p.57).
2.4
Small Area Estimation: Univariate Area
Level Models
Model-based SAE techniques can use mixed models with random effects. The ran-
dom effects explain the between-area variation which cannot be explained by the
auxiliary variables included in the model. Small area models can be defined at the
area or the unit level (Rao and Molina, 2015, p.76). Unit level small area models can
be developed only when auxiliary data are available for each population unit. How-
ever, such data are not available in many situations. In this case, area level small
area models, which relate small area direct estimates to the area level auxiliary data,
are applied (Jiang and Lahiri, 2006).
The basic area level model proposed by Fay and Herriot (1979) is often called
the Fay-Herriot (FH) model. It is a linear random effects model which links the
parameter of interest θd (say, a mean or proportion) in area d on the auxiliary
variables Zd= (Zd1, Zd2. . . , Zdp)
0
through the linking model:
θd=Zd0β+vd, d= 1,2, . . . , D (2.18)
whereθd=g( ¯Yd),βis a (p×1) vector of regression coefficients andvdare identically,
independently and normally distributed area-specific random effects with E(vd) =
0,Var(vd) =σv2. It is not necessary that Var(vd) are all equal.
The sampling model is:
ˆ
where ˆθd=g(Yb¯d) anded are independent sampling errors normally distributed with
E(ed) = 0, Var(ed) = ψd. The sampling model indicates how the sample estimates
are related to the unknown population values and sampling errors ed. In practice,
ˆ
θd is usually a model-assisted or design-based estimator of θd.
Combining the sampling and linking model, the typical form of the univariate
Fay-Herriot model is
ˆ
θd =Zd0β+vd+ed, d= 1,2, . . . , D (2.20)
The two error terms, vd and ed, are assumed to be independent of each other
within and across areas. Here, vd is the area-specific random effects (also called the
model error) which measures the heterogeneity among the areas after allowing for
the covariates in the model (Ghosh and Rao, 1994) andψdis a design-based sampling
variance. Usually, it is assumed that ψd is known. The variance component σv2 is
unknown and has to be estimated from data under model (2.20).
For known variance σ2
v, under model (2.20), the BLUP for θd is a weighted
average of the direct estimator ˆθdand the regression synthetic estimator Zd0βˆwhich
is ˆ θdBLU P = Zd0βˆ+γd ˆ θd−Zd0βˆ = γdθˆd+ (1−γd)Zd0βˆ (2.21) where γd = σ2 v ψd+σv2
and ˆ β= D X d=1 (ψd+σv2) −1Z dZd0 !−1 D X d=1 (ψd+σ2v) −1Z dθˆd ! (2.22)
(Rao and Molina, 2015, p.124). Since the Fay-Herriot model deals with the area
level quantities ˆθd and not with the individual observations, the BLUP estimators
are valid for general sampling designs (Rao, 2003b, p.116). When the model variance
σ2
v is relatively small, γd will be small and more weight is attached to the synthetic
estimator. On the other hand, the direct estimator is given more weight if the design
variance ψd is relatively small. Here, γd is also called a ‘shrinkage factor’ since it
‘shrinks’ the direct estimator towards the synthetic estimator Zd0βˆ in (2.16). The MSE of the BLUP estimates under the area level model (2.21) is
MSE(ˆθBLU Pd ) = E(ˆθBLU Pd −θd)2 =g1(σv2) +g2(σv2) (2.23)
(Rao and Molina, 2015, p.125) where
g1(σv2) = σ2 vψd ψd+σv2 =γdψd and g2(σv2) = (1−γd)2Zd0 " X d ZdZd0/(ψd+σv2) #−1 Zd
The first term,g1(σv2), in (2.23) isO(1) while, due to estimatingβ, the second term,
g2(σv2), is O(D
−1) for large D. It is obvious that when the variance of the model
error, vd, is small relative to the total variance, ˆθdBLU P is much more efficient than
ˆ
θd which has variance ψd (Rao and Molina, 2015, p.126).
The BLUP estimator (2.21) relies on the variance componentσ2
v which is usually
ponents in a linear mixed model are available including: the method of moments,
maximum likelihood (ML), and restricted maximum likelihood (REML) (Cham-
bers and Clark, 2012, p.168). An empirical BLUP or EBLUP estimator can then
be obtained by replacing σ2
v with a consistent estimator ˆσv2 under the frequentist
approach, or the empirical Bayes predictor under the Bayesian approach when as-
suming normality (Chambers and Clark, 2012, p.169). The latter predictor is the
posterior mean of θd, but with σv2 replaced by a sample estimate obtained from the
marginal distribution of the direct estimators given the variance. Alternatively, one
may compute the hierarchical Bayes predictor by assuming prior distributions for
β and σ2
v and computing the posterior distribution of θd given the available data
(e.g. Pfeffermann, 2013). In this thesis, frequentist approach is used because these
estimators are ML estimators, unlike the Bayesian estimators, they are independent
of the choice of priors.
An important matter of SAE is to determine the accuracy of the predictors. A
na¨ıve measure of uncertainty of an EBLUP is the MSE of the corresponding BLUP
which is also known as prediction MSE (PMSE) and can be defined as E(ˆθd−θd)2
(Pfeffermann, 2013). Prasad and Rao (1990) developed PMSE estimators of EBLUP
with bias of order o(D−1) under the linear mixed model (2.20). Prasad and Rao
(1990) assume that the random errors follow normal distribution and the model
variances are estimated by the ANOVA method of moments. Datta and Lahiri
(2000) extended this to the case where the variance components are estimated by
the ML or REML method.
model (2.20) as
MSE(ˆθdEBLU P) = E(ˆθdEBLU P −θd)2
= g1(σ2v) +g2(σv2) +g3(σ2v) (2.24) with g1(σv2) = σ2 vψd ψd+σv2 =γdψd g2(σv2) = (1−γd)2Zd0 " X d ZdZd0/(ψd+σv2) #−1 Zd g3(σv2) = ψ 2 d(ψd+σv2) −3var(ˆσ2 v)
An approximately unbiased estimator of MSE(ˆθEBLU P
d ) is
[
MSE(ˆθEBLU Pd ) = g1(ˆσv2) +g2(ˆσv2) + 2g3(ˆσv2) (2.25)
(Prasad and Rao, 1990). Estimation of the PMSE under generalized linear mixed
models (GLMM) is more complex and in such cases resampling procedures can be
used (e.g. Pfeffermann, 2013). Resampling procedures are usually accepted as a
good alternative to analytical approximations to the MSE of the EBLUP. They are
very attractive due to their conceptual simplicity and so can be applied to complex
statistical models (e.g. Molina et al., 2007). See for example Jiang et al. (2002),
who proposed a jackknife method in the context of small area estimation. Further
resampling methods for the small area framework are the parametric bootstrap
approaches of Gonz´alez-Manteiga et al.(2008) and Hall and Maiti (2006) as well as