How Nearly Can Model-Based Prediction and Design-Based Estimation Be Reconciled?

Abstract

Abstract Two general linear model-based predictors, one of the expectation of a finite population total and one of that total itself, are compared with the design-based generalized regression estimator (GRE). First, the predictors are made to conform to the GRE by modifying the regression parameter estimators but retaining the same (optimal) inclusion probabilities. Second, the GRE is made to conform with each of the predictors in turn by modifying the inclusion probabilities but retaining the generalized least squares (GLS) or best linear unbiased form for the estimators of the regression parameters. It is shown that the choice of inclusion probabilities is more important asymptotically than the choice of estimator for the regression parameters and hence that predictors obtained by the first method generally have smaller asymptotic expected variances than those obtained by the second method. Using the first method, certain special cases are shown to correspond to familiar estimators. If there is only one explanatory variable and no constant term in the model, the predictor of the expectation of the finite population total obtained by the first procedure is identical to the Horvitz-Thompson ratio estimator. If the finite population total itself is to be predicted, the estimator is that suggested by Brewer (1979). If there is a constant term in the model, the GLS predictor can be conformed to the GRE by replacing the inverse-variance weights used to estimate the regression coefficients by functions of the (optimal) inclusion probabilities. It is shown further that appropriate estimators can be constructed in the general case by the use of an appropriate instrumental variable. Under fairly weak conditions this variable can be constructed by deleting a column from a matrix used in the calculation of the GLS estimator and replacing it by a column of weights that are simple functions of the inclusion probabilities. Illustrative examples are given.