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The variances can be estimated using (S¨rndal et al. 1992 , p. 246, 248, and 223)
g l , RAT ^ l
XX k , l2s Δ
¼
^ kl g k , RAT ^ k
V t RAT , y
;
ð
10
:
3
Þ
XX k , l2s Δ
¼
^ kl ^ k ^ l ;
V t DIF , y
ð
10
:
4
Þ
where ^ k ¼
y k ¼ Bx k , are the expanded errors that
assume x and y are proportional. The coefficient
e k k ¼
ð
y k ^
y k
Þ=π k with
^
B
¼ t HT , y =t HT , x is the ratio of
the two HT estimators.
The corrections induced by these estimators reduce the uncertainty as much as
their basic hypothesis (proportional or additive errors) is respected by the sample
data. If every unit respects this proportional or additive errors hypothesis, the
variance of the estimates will be 0.
The main drawback of this approach is that it is based on a univariate framework.
However, its extension is implicit in Eq. ( 10.3 ), where a simple linear regression
model y k ¼ Bx k þ
e k has already been assumed.
Now, let us assume that q auxiliary variables are available. Define a
q dimensional vector of auxiliary variables associated with unit k as
x k ¼
t , k
, N . The totals of the q auxiliary variables
are known for the population units. So, for each element k
ð
x k 1
x k 2
...
x kq
Þ
¼
1,
...
s , we observe ( y k , x k ).
The univariate approach can be extended to a multiple linear regression model
y k ¼
2
x k B
) to allow any number of covariates to be included in
the model. By assuming this multivariate regression model, we obtain the general-
ized regression estimator (GREG) defined as (S¨rndal et al. 1992 , p. 231)
þ
e k (denoted as
ʾ
X k2U ^
X k2s ^ k ¼
X k2U ^
X k2s d k y k
;
x k B
t GREG , y ¼
y k þ
y k þ
ð
10
:
5
Þ
where the regression parameters are estimated using
1
X k2s
X k2s
x k x k
˃
x k y k
˃
B
¼ T 1
t
;
ð
:
Þ
¼
10
6
k
π k
k
π k
X k2s
X k2s
x k x k
˃
x k y k
˃
with T
π k , t
k arises from the introduction of the
¼
¼
π k and the
˃
k
k
linear multivariate regression model
ʾ
, and in particular represents the variance of
y k under the model
.
Denote the vector of known totals for the q auxiliary variables as t x , and the
vector of their HT estimates as t HT , x . Then, it is interesting to note that Eq. ( 10.5 )
can also be written as
ʾ
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