Agriculture Reference
In-Depth Information
Þ
1
u
˅
¼
ˁ
˅þ
u
) ˅
¼ I
ˁ
ð
;
ð
11
:
41
Þ
W
W
where u is a
d
1 vector of independent error terms with zero mean and constant
variance
2
u
, and I is the usual
dd
identity matrix. Substituting Eq. (
11.41
) into
Eq. (
11.29
), it is possible to define
˃
ʸ
¼X
Þ
1
u
þ
e
βþ
BI
ˁ
ð
;
ð
11
:
42
Þ
W
˅
where e is independent of
. Model (
11.42
) considers spatially correlated random
area effects. In this case,
˅
has covariance matrix G such that
h
i
Þ
1
Þ
1
2
u
W
t
G¼
˃
ð
I
ˁ
W
ð
I
ˁ
;
ð
11
:
43
Þ
where I
ˁ
ð
W
Þ
is non-singular, and e has covariance matrix
R ¼
ˈ
¼
diag
ˈð :
ð
11
:
44
Þ
, the covariance matrix of
ʸ
Because e is independent of
˅
can be defined as
h
i
B
t
Þ
1
Þ
1
V ¼ R
þ
BGB
t
2
I
ˁ
W
t
¼
diag
ˈðÞþ˃
u
BI
ˁ
W
ð
ð
:
ð
11
:
45
Þ
The spatial BLUP estimator of
ʸ
d
is (Pratesi and Salvati
2008
)
n
h
i
o
B
t
ʸ
d
,
SBLUP
¼ x
d
βþ
z
d
Þ
1
Þ
1
2
u
W
t
˃
ð
I
ˁ
W
ð
I
ˁ
n
h
i
B
t
o
1
;
ð
11
:
46
Þ
Þ
1
ʸ
X
β
Þ
1
u
BI
ˁ
W
t
x
diag
ˈðÞþ˃
ð
W
ð
I
ˁ
1
X
t
V
1
where
β
¼ X
t
V
1
X
ʸ
, and z
t
d
is the 1
d
vector (0,0,
,0) (with
1 in the
d
-th position). Obviously the spatial BLUP reduces to the traditional BLUP
when
...
,0,1,0,
...
ˁ
¼0.
The spatial BLUP depends on the unknown variance
˃
u
and
ˁ
. Replacing these
parameters with their corresponding estimators, we can define a two-stage estimator
called spatial EBLUP (SEBLUP)
n
h
i
o
B
t
ʸ
d
,
SEBLUP
¼ x
d
βþ
z
d
Þ
1
Þ
1
W
t
u
˃
ð
I
ˁ
W
ð
I
ˁ
n
h
i
B
t
o
1
ð
11
:
47
Þ
Þ
1
Þ
1
ʸ
X
β
W
t
x
diag
ˈðÞþ˃
u
BI
ˁ
ð
W
ð
I
ˁ
:
2
u
Assuming that the random effects are normally distributed,
can be
estimated using either ML or REML procedures. For further details about the
estimation procedure, the MSE, and the estimate of the MSE that can be obtained
analogously to the EBLUP estimator, see Pratesi and Salvati (
2008
).
˃
and
ˁ
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