i DESIGN BASED PREDICTION IN SIMPLE RANDOM SAMPLING WITH APPLICATION TO RANDOM EFFECTS

We develop estimators of unit parameters using a prediction approach in an expanded permutation model whose basis is a simple random sampling design. The unit parameters are defined as linear combinations of the expanded random variables. We distinguish these parameters from other linear combinations of random variables that are typically defined as random effects. In the context of the expanded model, we illustrate that the linear combinations that define unit parameters and random effects differ, as do their estimators. The unbiased minimum variance estimator of the 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. The expanded permutation model provides a simple probability framework that unifies inference for problems where separate design based and model based frameworks have been used. It also suggests C02ed04.doc 27/03/02 8:06 (formerly srsv26.doc) ii extensions to other higher dimension models defined via additional random variables that may result in improved estimators