Calibration Estimators in Survey Sampling

Abstract

This article investigates estimation of finite population totals in the presence of univariate or multivariate auxiliary information. Estimation is equivalent to attaching weights to the survey data. We focus attention on the several weighting systems that can be associated with a given amount of auxiliary information and derive a weighting system with the aid of a distance measure and a set of calibration equations. We briefly mention an application to the case in which the information consists of known marginal counts in a two- or multi-way table, known as generalized raking. The general regression estimator (GREG) was conceived with multivariate auxiliary information in mind. Ordinarily, this estimator is justified by a regression relationship between the study variable y and the auxiliary vector x. But we note that the GREG can be derived by a different route by focusing instead on the weights. The ordinary sampling weights of the kth observation is 1/πk , where πk is the inclusion probability of k. We show that the weights implied by the GREG are as close as possible, according to a given distance measure, to the 1/πk while respecting side conditions called calibration equations. These state that the sample sum of the weighted auxiliary variable values must equal the known population total for that auxiliary variable. That is, the calibrated weights must give perfect estimates when applied to each auxiliary variable. That is a consistency check that appeals to many practitioners, because a strong correlation between the auxiliary variables and the study variable means that the weights that perform well for the auxiliary variable also should perform well for the study variable. The GREG uses the auxiliary information efficiently, so the estimates are precise; however, the individual weights are not always without reproach. For example, negative weights can occur, and in some applications this does not make sense. It is natural to seek the root of the dissatisfaction in the underlying distance measure. Consequently, we allow alternative distance measures that satisfy only a set of minimal requirements. Each distance measure leads, via the calibration equations, to a specific weighting system and thereby to a new estimator. These estimators form a family of calibration estimators. We show that the GREG is a first approximation to all other members of the family; all are asymptotically equivalent to the GREG, and the variance estimator already known for the GREG is recommended for use in any other member of the family. Numerical features of the weights and ease of computation become more than anything else the bases for choosing between the estimators. The reasoning is applied to calibration on known marginals of a two-way frequency table. Our family of distance measures leads in this case to a family of generalized raking procedures, of which classical raking ratio is one.