A Functional Form Approach to Calibration
Victor M. Estevao and Carl-Erik Särndal
Calibration has become a widely used procedure for estimation in sample surveys.
It uses auxiliary information to produce efficient estimates. Calibration requires
that we know population totals (control totals) for one or more auxiliary
variables (x-variables). The efficiency of the calibration estimator
depends on how well the auxiliary variables explain the variability of y,
the variable of interest. Traditionally, a distance minimization approach
is used for building calibration estimators. The distance measures that have
been proposed produce estimators that are nearly identical, so this approach
does not provide much insight into the properties of different calibration estimators.
In this article, we note that distance minimization is not the only possible
starting point for calibration. We define and develop an alternative, the functional
form approach. The calibrated weights are given a simple mathematical
form that depends on two parameters. This defines a family of calibration
estimators denoted by CALF.
It includes the family of generalized regression (GREG) estimators GREG. We discuss the role of
the auxiliary variables in the calibration. To do this, we assume a linear relation
between y and the x-variables. In most surveys, the x-variables
in the calibration are not the only ones that explain y. In the unlikely
event that they do (except for random noise), we say that the model is saturated
by the calibration. This case is not generally of interest because the resulting
estimators have similar properties. Usually, other x-variables are
significant in explaining y but are excluded from the calibration because
they are either not observed in the sample or their control totals are unknown.
In this case, the model is called unsaturated. We look at the unsaturated
model and show that for some sample designs, we can find a GREG
which has minimum Taylor variance among the estimators in CALF. The Monte Carlo simulations at the
end of the article illustrate these results.
Calibration estimators; GREG estimator; calibrated weights; auxiliary information.