Replication variance estimation after sample-based calibration
Section 3. Sample-based raking calibration
In the application to the two 2016 FHWAR surveys, we
used raking rather than regression estimation for calibration. As for
regression estimation, the goal is to create new raking controls for each of
the replicates, so that the replicate variance estimator for the primary survey
accounts for the variability of the control totals from the secondary survey.
The above results for regression estimation do not apply directly, but we can
apply the same reasoning as in Deville and Särndal (1992) to show that they
continue to hold for raking. Instead of relying on this general result, we will
derive it here explicitly to show how to obtain the adjusted control totals for
the replicates.
For simplicity, we describe here the case in which we
are controlling for the marginal counts in domains defined by the levels of 2
categorical variables, denoted
and
having
and
levels, respectively. In the primary survey,
the estimated counts in the domains defined by the intersections of the two
variables are
with
if element
is in the domain defined by the intersection
of
and
and 0 otherwise. The marginal estimated counts
are defined analogously,
and
We write
for the vector of indicators for the marginal
domains for element
The estimated marginal counts in the primary
survey are
and the corresponding control totals from the
secondary survey are
Using the classical raking ratio algorithm of Deming
and Stephan (1940) until convergence, the raked weights for the primary survey
can be written as
where
is a solution to the system of
equations
The solution to these equations is not unique, so
one of the unknowns can be set to 0 and an equation removed. This does not
affect the values of
and we will set
and remove the last equation in what follows.
There is no explicit expression for the solution to (3.2),
but it can be approximated by using a linearization argument. Under the usual
survey asymptotic framework that ensures design consistency of Horvitz-Thompson
estimators, the
and
converge to 0 as the sample sizes of the two
surveys increase, so that expansion around 0 is valid. Doing so for the
equations in (3.2), we approximate the reduced set of
equations by
which can be rewritten as
Ignoring the smaller order remainders, the solution
to this system of linear equations can be written in the form
where
is a symmetric
matrix containing estimated domain counts
readily obtained from the left-hand side of (3.3). Note that the resulting
is not a linear estimator, because in the
linearization we conditioned on
Finally, plugging
into the expression (3.1) and linearizing
again, we obtain
and the estimator after raking is
Note that the size of the control variable and associated
estimates is now but we maintain the
prior notation for simplicity. In (3.5), corresponds to in the regression
estimator (2.2). This asymptotic equivalence between the raking estimator and
the regression estimator with the same control totals was established by Deville
and Särndal (1992). In particular, they provide sufficient conditions under
which the asymptotic variance of the equivalent regression estimator can also
be used for inference for the raking estimator. Hence, a variance estimator of
the form (2.6) can be constructed, with
We now consider the construction of replicate weights
for the primary survey that estimate the asymptotic variance of the raking
estimator. As before, we construct new replicate control totals using the replicate
estimates from the secondary
survey. Each of the sets of replicate weights of the primary survey
are adjusted by raking to its corresponding set of control totals, to obtain
the and the raked
replicate estimates Using the
approximation in (3.5) for each replicate, we obtain that the resulting
replication variance estimator is consistent for the asymptotic variance of the
raking estimator.
ISSN : 1492-0921
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