Likelihood Functions in Finite Population Sampling Theory
Richard M. Royall · Biometrika · 1976
Both conventional randomization models and prediction, i.e. superpopulation, models for finite populations are discussed from the viewpoint of the likelihood axiom. It is argued that finite populations do fall within the scope of this axiom, and that the likelihood function, when properly defined and interpreted, can play the same fundamental role in finite population theory as it does elsewhere in statistical inference. Under a multivariate normal regression model, the relationship between the likelihood function for the population total and the probability distribution of the minimum variance unbiased estimator is studied. The anticipated result, that this estimator maximizes the likelihood function, is shown to hold under the most familiar models, but an example shows it to be false in general. The role of balanced samples in providing robust inferences is discussed briefly. Conditions are described under which the likelihood function is unchanged by the addition of a new regressor to the model.