Model Quality Report in Business Statistics - Harvard ...

In the case of poststratification the auxiliary vector is defined by x = ( γ γ γ ) T
k
1 k
,...,
pk
,...,
Pk
⎧1 if unit k ∈U
p
2
where, for p = 1,..., P,
γ
pk
= ⎨
and σ k
= 1 for all k. This poststratification
⎩0
otherwise
approach gives us one simple method of dealing with outlying observations in a survey, since
they can be moved into an appropriate poststratum for estimation.
Most of the classical estimators can be derived as special cases from the GREG estimator.
For example, if x
k
= xk
for all k and σ 2 k
∝ x k
, where x k
is a continuous variable, and when
nonresponse model (i) is used, then the following estimator is obtained:
$t
yr
=
H
∑ N y
h=
1
H
h
∑ N x
h=
1
h
rh
rh
∑
U
x
k
(2.13)
Estimator (2.13) is sometimes called the combined ratio estimator.
Sometimes the group totals ∑U
p
x
k
are known and, in this general case, the p-groups are
called model groups. Let us present a simple example. As before assume that x k
is a
continuous variable, but here we know the quantities ∑ ; p = 1,..., P. Let
( γ x γ x γ x ) T
x
k
=
1 k k
,...,
pk k
,...,
Pk k
, σ 2 k
∝ x k
for each p-group, and the RHGs coincide with
strata (nonresponse model (i)) then the GREG estimator takes the form
U p
x k
t$
yr
=
H
$
P
∑ N
h=
1
∑
H
p=
1
∑ N$
h=
1
hp
hp
y
x
rhp
rhp
∑
U p
x
k
(2.14)
If strata and model groups coincide then estimator (2.14) can be written
$t
yr
H yr
h
= ∑ ∑
h=1
x
r
h
U
h
x
k
(2.15)
Estimator (2.15) is sometimes called the separate ratio estimator.
When x
k
= ( 1, xk
) for all k and σ
2 k
= constant , then the classical regression estimator is
obtained.
Many business surveys are subject to occasional unusual observations, or outliers, which can
have a large effect on the estimates. In these cases, robust versions of point estimators are
often used, with the simplest being the poststratification estimator with the outliers in their
own (completely enumerated) poststratum. This follows from the method above (2.13). Other
methods involve adjusting the weights or the responding values, and winsorisation is
becoming widely used within the UK for treating outliers. This leads to a different estimator,
which does not necessarily fit completely into the GREG framework.
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