Variance estimation under monotone non-response for a panel survey
Section 3. Calibration and complex parameters
In most surveys, a calibration step is used to obtain
adjusted weights which enable to improve the accuracy of total estimates. Such
calibrated estimators are considered in Section 3.1. Also, more complex
parameters than totals are frequently of interest, and a linearization step can
be used for variance estimation. This is the purpose of Section 3.2. The
estimation of complex parameters with calibrated weights is treated in Section 3.3.
In each case, explicit formulas for variance estimation and simplified variance
estimation are derived, and the bias of the simplified variance estimator is
discussed.
3.1 Variance estimation for calibrated total
estimators
Assume that a vector
of auxiliary variables is available for any
unit
and that the vector of totals
on the population
is known. Then an additional calibration step (Deville
and Särndal, 1992) is usually applied to
It consists in modifying the weights
to obtain calibrated weights
which enable to match the real total
in the sense that
The
new calibrated weights are chosen to minimize a distance function with the
original weights, while satisfying (3.1). This leads to the calibrated estimator
The estimated residual for the weighted regression of
on
is denoted by
with
Replacing in (2.11) the variable
with
yields the estimator of the variance due to
the sampling design
Similarly, replacing in (2.12) the variable
with
yields the estimator of the variance due to
the non-response
The global variance estimator for
is
The simplified estimator of the variance due to
non-response is
Here again, this simplified variance estimator ignores the prediction
terms
If the underlying calibration model is
appropriate, then the explanatory power of
for
is expected to be small, as well as the bias
of the simplified variance estimator. On the other hand, if there remains in
some significant part of
that may not been explained by
the bias of the simplified variance estimator
may be non-negligible. This may occur in case of domain estimation, when the
calibration variables do not include any auxiliary information specific of the
domain.
3.2 Variance estimation for complex parameters
We may be interested in estimating more complex
parameters than totals. Suppose that the variable of interest
is
multivariate, and that the parameter of
interest is
with
a known function. At time
substituting
into
yields the plug-in estimator
The estimated linearized variable of
is
with
the
vector of first derivatives of
at point
Replacing in (2.11) the variable
with
yields the estimator of the variance due to
the sampling design
Similarly, replacing in (2.12) the variable
with
yields the estimator of the variance due to
the non-response
The global variance estimator for
is
The simplified estimator of the variance due to
non-response is
The bias of this simplified variance estimator will depend on the
explanatory power for
on the linearized variable
3.3 Variance estimation for complex parameters
under calibration
The calibrated weights
may be used
to obtain an estimator of the parameter
Substituting
into
yields the
calibrated plug-in estimator
To obtain a variance estimator for
we first
compute the estimated linearized variable
and take
with
Replacing in (2.11) the variable
with
yields the estimator of the variance due to
the sampling design
Similarly, replacing in (2.12) the variable
with
yields the estimator of the variance due to
the non-response
The global variance estimator for
is
The simplified estimator of the variance due to
non-response is
Since the variable
is obtained as the residual in the regression
of the linearized variable
on the calibration variables
the explanatory power for
on
is expected to be small in practice, and the
bias of the simplified variance estimator is expected to be small as well.
ISSN : 1492-0921
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