Optimal sampling and estimation strategies under the linear model

In some cases model-based and model-assisted inferences can lead to very different estimators. These two paradigms are not so different if we search for an optimal strategy rather than just an optimal estimator, a strategy being a pair composed of a sampling design and an estimator. We show that, under a linear model, the optimal model-assisted strategy consists of a balanced sampling design with inclusion probabilities that are proportional to the standard deviations of the errors of the model and the Horvitz--Thompson estimator. If the heteroscedasticity of the model is ‚fully explainable’ by the auxiliary variables, then this strategy is also optimal in a model-based sense. Moreover, under balanced sampling and with inclusion probabilities that are proportional to the standard deviation of the model, the best linear unbiased estimator and the Horvitz--Thompson estimator are equal. Finally, it is possible to construct a single estimator for both the design and model variance. The inference can thus be valid under the sampling design and under the model. Copyright 2008, Oxford University Press.