On the consistency between model-and design-based estimators in survey sampling

In finite population sampling, often a distinction is made between model-and design-based estimators of the parameters of interest (like the population total, population variance, etc.). The model-based estimators depend on the (known) parameters of the model, while the design-based estimators depend on the (known) selection probabilities of the different units in the population. It is shown in this paper that the two approaches are not necessarily incompatible, and indeed can often lead to the same estimator. Our ideas are illustrated with the Horvitz-Thompson, and the generalized Horvitz-Thompson estimator. These estimators are identified as hierarchical Bays estimators. Also, certain “stepwise-Bayes” estimators of Vardeman and Meeden (J. Stat. Inf. (1983), V7, pp 329-341) are unified from a hierarchical Bayes point of view.