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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