Agriculture Reference
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ʸ d ¼ ʸ d þ e d , d ¼ 1, 2,
...
, D
;
ð 11
:
27 Þ
iid
where e d ʸ d
j
ð
0
; ˈ d
Þ
are the sampling errors. This assumption implies that the
e
ʸ d are not biased with respect to the design. Besides, the samples
estimators
variances
ˈ d should be known. The latter hypothesis can be quite restrictive in
some applications, and it should be relaxed (Ghosh and Rao 1994 ).
Combining Eqs. ( 11.26 ) and ( 11.27 ), we obtain the model
ʸ d ¼ x d β þ b d ˅ d þ e d , d ¼ 1, 2,
...
, D
:
ð 11
:
28 Þ
The model in Eq. ( 11.28 ) is known as the Fay-Herriott model (1979), and repre-
sents a mixed linear model. Note that Model ( 11.28 ) involves two random compo-
nents: the first ( e d ) caused by the design and the second ( ˅ d ) caused by the model.
In matrix notation, Model ( 11.28 ) can be expressed as
ʸ ¼ X β þ B ˅ þ e ;
ð 11 : 29 Þ
where ʸ
is the d 1 vector of the direct estimators, X is the dq matrix of auxiliary
variables,
the q1 regression parameters vector, B is the diagonal matrix of
order d of known constants,
β
is the d 1 vector of random component area specific
effects, and e is the d 1 vector of the sampling errors. Note that Model ( 11.29 )is
the same model used in Sect. 7.5 .
If
˅
ʸ d is non-linear function of the total t d , and the sample size n d is small, the
assumption Ee d ʸ d
jð Þ ¼0 may not be valid, even if the direct estimator t d is design-
unbiased. In this case, a more appropriate model for the sampling errors is
t d ¼ t d þ e d
Ee d
¼ 0
:
ð 11
:
30 Þ
j
t d
It is evident that Model ( 11.30 ) cannot be combined with Model ( 11.26 ), so the
usual procedures and results of linear mixed model theory do not apply. For some
suggestions about the solution of this problem see Rao ( 1999 ).
Model ( 11.28 ) has been extensively analyzed by many researchers, and various
extensions have been proposed. Fay ( 1987 ) and Datta et al. ( 1991 ) proposed the
multivariate version of the Fay-Herriot model ( 11.27 ), and proved that this
approach can lead to more efficient estimators. Isaki et al. ( 2000 ) used area level
models that accounted for correlated sampling errors. Rao and Yu ( 1994 ) suggested
an extension of Model
( 11.28 )
for
the analysis of
time series and cross-
sectional data.
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