Optimal sampling and estimation strategies under the linear model

In some cases model-based and model-assisted inferences canlead to very different estimators. These two paradigms are notso different if we search for an optimal strategy rather thanjust an optimal estimator, a strategy being a pair composedof a sampling design and an estimator. We show that, under alinear model, the optimal model-assisted strategy consists ofa balanced sampling design with inclusion probabilities thatare proportional to the standard deviations of the errors ofthe model and the Horvitz–Thompson estimator. If the heteroscedasticityof 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 thatare proportional to the standard deviation of the model, thebest linear unbiased estimator and the Horvitz–Thompsonestimator are equal. Finally, it is possible to construct asingle estimator for both the design and model variance. Theinference can thus be valid under the sampling design and underthe model.