Agriculture Reference
In-Depth Information
direct estimators can be defined. This approach includes the HT direct estimator and
the GREG estimator for small domains (see Sect.
10.2
).
If
N
d
is unknown, the simplest direct estimator of the total for the
d-
th SA can be
expressed as (compare with Sect.
1.2
)
dir
t
d
¼
X
s
d
^
k
¼
X
s
d
ð
y
k
=π
k
Þ:
ð
11
:
6
Þ
The estimator in Eq. (
11.6
) is referred to as the HT direct estimator for small
domains. When
N
d
is known, the ordinarily preferred estimator is
dir
t
d
¼
N
d
e
y
s
d
;
ð
11
:
7
Þ
where
X
s
d
^
k
X
s
d
y
k
=π
k
X
s
d
X
s
d
e
y
s
d
¼
Þ
¼
:
ð
11
:
8
Þ
ð
1
=π
k
ð
1
=π
k
Þ
However, whether
N
d
is known or not, the estimator in Eq. (
11.8
) (i.e.,
y
d
¼
e
y
s
d
)is
highly recommended for estimating the mean (S¨rndal et al.
1992
, p. 391).
The variance of the estimator in Eq. (
11.6
) is (compare with Sect.
1.2
)
¼
XX
U
d
Δ
kl
^
k
^
l
;
V
dir
t
d
ð
11
:
9
Þ
and a design-unbiased variance estimator for Eq. (
11.9
)is
¼
XX
s
d
Δ
^
kl
^
k
^
l
,
V
dir
t
d
ð
11
:
10
Þ
^
kl
¼
Δ
kl
=π
kl
:
where
^
k
¼
y
k
=π
k
and
Conversely, the variance of the estimator in
Eq. (
11.7
) is approximately (S¨rndal et al.
1992
,p.391)
Δ
y
l
y
U
d
π
l
XX
U
d
Δ
kl
y
k
y
U
d
π
k
V
dir
t
d
;
ð
11
:
11
Þ
and its variance estimator is
2
y
l
e
XX
s
d
Δ
¼
y
k
e
N
d
N
d
y
s
d
y
s
d
^
kl
V
dir
t
d
;
ð
11
:
12
Þ
π
k
π
l
where
N
d
¼
X
s
d
=πð Þ:
It is worth noting that the HT direct estimator is not generally designed to use
auxiliary information or covariates. When
1
π
k
>
0 and there are sufficiently many
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