A prediction approach to sampling design

Standard approaches to sample surveys take as the point of departure the estimation of one or several population totals (or means), or a few predefined sub-totals (or sub-means). While the model-based prediction approach provides an attractive framework for estimation and inference, a model-based theory for the variety of randomization sampling designs has been lacking. In this paper we extend the model-based approach to the prediction of individuals in addition to totals and means. Since, given the sample, the conditional prediction error is zero for the selected units but positive for the units outside of the sample, it is possible to use the sampling design to control the unconditional individual prediction mean square errors. This immediately raises the need for probability sampling. It turns out that balancing between optimal prediction of the population total and control over individual predictions provides a fruitful model-based approach to sampling design. Apart from raising the need for probability sampling in general, it leads naturally to a number of important design features that are firmly established in the sampling practice, including the use of simple random sampling for homogeneous populations and unequal probability sampling otherwise, the division of a business population into the take-all, take-some and take-none units, the most common two-stage sampling designs, the use of stratification with proportional allocation, etc.. Most of them have not received adequate model-based treatment previously. Our approach enables us to give an appraisal of these methods from a prediction point of view.