Agriculture Reference
In-Depth Information
2. Given a sample
s
and the function
F
(.), solve the calibration equations in
Eq. (
10.19
) for
, where the last quantities of Eq. (
10.19
) are known.
3. Compute the calibration estimator of
t
y
according to Eq. (
10.20
).
This estimator will give more accurate estimates of
t
y
as the relationship between
X and y strengthens. The following examples of distance functions (
G
) were given
by Deville and S
¨
rndal (
1992
):
λ
2
1. Chi-squared distance:
w
k
,
CAL
ð
d
k
Þ
=
2
d
k
˄
k
.
˄
k
1
w
k
,
CAL
log
w
k
,
CAL
=
2. Logarithm distance:
ð
ð
d
k
Þ
w
k
,
CAL
þ
d
k
Þ
.
p
d
k
2
p
w
k
,
CAL
3. Hellinger distance: 2
=˄
k
.
˄
k
1
4. Minimum entropy distance:
ð
d
k
log
w
k
,
CAL
=
ð
d
k
Þ þ
w
k
,
CAL
d
k
Þ
.
2
5. Modified chi-squared distance:
w
k
,
CAL
ð
d
k
Þ
=
ð
2
w
k
,
CAL
˄
k
Þ
.
6. Truncated (
L
,
U
) logarithm distance or Logit:
d
k
log
log
1
A
w
k
,
CAL
d
k
w
k
,
CAL
=
L
w
k
,
CAL
d
k
U
ð
w
k
,
CAL
=
d
k
Þ
L
þ
U
:
ð
1
L
Þ
U
1
ð
10
:
21
Þ
w
k
,
CAL
d
k
when
L
U
,if
L
is large negative and
U
is large positive, we are close to
0 and
U
is large we are close to distance 2.
7. Truncated (
L
,
U
) chi-square distance:
w
k
,
CAL
distance 1. If
L
¼
w
k
,
CAL
d
k
2
ð
d
k
Þ
=
2
d
k
˄
k
when
L
U
,
and
1
otherwise.
˄
k
is a parameter that can be tuned to achieve the minimum, and
L
and
U
are two constants such that
L
Here,
<
<
U
and
A
¼
ð
U
L
Þ=
½
ð
L
Þ
ð
U
Þ
.
The choice of distance function depends on the requirements and peculiarities of
each estimation problem.
This general framework is important because it allows us to show that most
traditional estimators are a special case of the calibration estimator.
For example, the GREG estimator in Eq. (
10.5
) is a special case of the calibra-
tion estimator that uses the chi-square distance.
If we use this distance function, then
Fu
1
1
1
ðÞ
¼
ð
1
þ ˄
k
u
Þ
leads to the calibration
X
k2s
d
k
˄
k
x
k
x
k
(assuming that T
1
s
exists), and t
HT
,
x
is the HT estimator for x. The resulting
calibration estimator is the same as the GREG estimator in Eq. (
10.5
). However, if
we again consider the chi-squared distance function but with
, where
, T
s
¼
t
x
t
HT
,
x
þ ˄
k
x
k
λ
T
1
s
weight
w
k
,
CAL
¼
d
k
1
λ
¼
˄
k
¼
1
=
x
k
, we obtain
the ratio estimator in Eq. (
10.1
).
Deville et al. (
1993
), Zhang (
2000
), and Breidt and Opsomer (
2008
) explained
that the post-stratified estimator and
raking
are special cases of calibration estima-
tion, when the available information consists of known cell counts or known
marginal counts in a contingency table.
For simplicity, consider a two-way contingency table with
RW
rows,
CL
col-
umns, and
RW
CL
¼
q
cells. The generic cell (
rw
,
cl
) contains
N
rw,cl
elements.
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