A Functional Form Approach to Calibration

Victor M. Estevao and Carl-Erik Särndal

Abstract:

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.

Keywords:
Calibration estimators; GREG estimator; calibrated weights; auxiliary information.