Small-Area Estimation Under Informative Probability Sampling of Areas and Within the Selected Areas

Abstract

In this article we show how to predict small-area means and obtain valid mean squared error estimators and confidence intervals when the areas represented in the sample are sampled with unequal probabilities possibly related to the true (unknown) area means and the sampling of units within the selected areas is with probabilities possibly related to the outcome values. Ignoring the effects of the sampling process on the distribution of the observed outcomes in such cases may bias the inference very severely. Classical design-based inference that uses the randomization distribution of probability-weighted estimators cannot be applied for predicting the means of nonsampled areas. We propose simple test statistics for testing the informativeness of the selection of areas and sampling of units within the selected areas. We illustrate the proposed procedures by a simulation study and a real application of estimating mean body mass index in U.S. counties, using data from the Third National Health and Nutrition Examination Survey.