Model-assisted sample design is minimax for model-based prediction

Probability sampling designs are sometimes used in conjunction with model-based predictors of finite population quantities. These designs should minimize the anticipated variance (AV), which is the variance over both the superpopulation and sampling processes, of the predictor of interest. The AV-optimal design is well known for model-assisted estimators which attain the Godambe-Joshi lower bound for the AV of design-unbiased estimators. However, no optimal probability designs have been found for model-based prediction, except under conditions such that the model-based and model-assisted estimators coincide; these cases can be limiting. This paper shows that the Godambe-Joshi lower bound is an upper bound for the AV of the best linear unbiased estimator of a population total, where the upper bound is over the space of all covariate sets. Therefore model-assisted optimal designs are a sensible choice for model-based prediction when there is uncertainty about the form of the final model, as there often would be prior to conducting the survey. Simulations confirm the result over a range of scenarios, including when the relationship between the target and auxiliary variables is nonlinear and modeled using splines. The AV is lowest relative to the bound when an important design variable is not associated with the target variable.