Agriculture Reference
In-Depth Information
F
kl
þ
1
M þ
1
,
k 6
¼
l 2 U
π
kl
¼
;
ð
7
:
21
Þ
where
F
kl
denotes the number of times that units
k
and
l
jointly appear in the
M
samples. Using the same considerations as the estimator of the total, an asymp-
totically unbiased estimator of the variance
V
HT
,
M
t
HT
,
M
ð
Þ
is obtained by substitu-
ting
π
kl
in Eqs. (1.27) or (1.30).
When the asymptotic equivalence of the estimator has been established, we still
need to define how many samples (
M)
we need to guarantee a sufficient approxi-
mation of the HT estimators.
We can use the quantity
LðÞ
¼
MSE t
HT
,
M
π
kl
with
ÞV t
H
ðÞ
V t
H
ðÞ
as a relative
j
ð
j=
index of efficiency. Fattorini (
2006
) showed that
"
#
9
M þ
2
1
CV t
H
ðÞ
LðÞ
1
þ
;
ð
7
:
22
Þ
ð
Þπ
0
2
f
g
1
=
2
π
k
and
CV t
H
ðÞ
¼
V t
H
ðÞ
where
π
0
¼
min
f
g
=
t
. Moreover, they showed that the
absolute relative bias (ARB) of
t
HT
,
M
is
E t
HT
,
M
j
ð
Þt
j
1
M þ
2
ARB t
HT
,
M
ð
Þ
¼
Þπ
0
:
ð
7
:
23
Þ
t
ð
If
V
HT
,
M
t
HT
,
ð Þ
is higher than the HT variance, it is reasonable to assume that there
is additional uncertainty due to the estimation of
π
kl
. It follows that
1
ARB V
HT
,
M
t
HT
,
M
ð
Þ
2
;
ð
7
:
24
Þ
Þπ
00
CV t
H
ðÞ
ð
M þ
2
f
g
where
π
kl
. These expressions can be used to fix upper bounds for the loss
in efficiency and the bias of
t
HT
,
M
, as functions of the computational effort needed
in terms of
M
. The main drawback of this approach is that if we assume extremely
precautionary values for
CV t
H
ðÞ
and
π
00
¼ min
π
0
, billions of sample replications are needed
to ensure the required approximations, which would impose prohibitive costs. An
even greater effort is typically needed to bound Eq. (
7.24
), particularly if some
second-order probabilities are very small. For these reasons, an adaptive algorithm
has been developed to speed up the update of
M
using only the results of previous
sample selections (Fattorini
2009
).
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