A unified approach to estimation and prediction under simple random sampling

Abstract

We consider a probability model where the design-based approach to inference under simple random sampling of a finite population encompasses a simple random permutation super-population model. The model consists of an expanded set of random variables following a random permutation probability distribution that keeps track of both the units’ labels and positions in the permutation. In particular, since we keep track of the labels, the model allows us to attack the problem of estimation of a unit's parameter. While some linear combinations of the expanded set of random variables correspond to linear combinations of the unit parameters, other linear combinations correspond to random variables known as random effects. Using a prediction technique similar to that employed under the model-based approach, we develop optimum estimators of the linear combinations of the unit parameters and optimum predictors of the random effects. The unbiased minimum variance estimator of the population mean is the sample mean and of a unit parameter is the Horvitz–Thompson estimator if the unit is included in the sample, and zero otherwise. The predictor of the random variable at a given position in the permutation is the realized unit's parameter for positions in the sample, and the sample mean for other positions. For other linear functions, unique minimum variance unbiased estimators may not exist.