Two-stage sampling from a prediction point of view when the cluster sizes are unknown

• Published in: Biometrika 2008-03, Vol.95 (1), p.187-204
• Main Authors: Bjørnstad, Jan F., Ytterstad, Elinor
• Format: Article
• Language: English
• Subjects: Applications; Biology, psychology, social sciences; Estimating techniques; Estimators; Exact sciences and technology; General topics; Law of likelihood; Mathematics; Modeling; Musical intervals; Optimal predictor; Optimization; Parameter estimation; Population; Population model; Prediction interval; Predictions; Predictive likelihood; Predictive modeling; Probability and statistics; Probability theory and stochastic processes; Random variables; Sample size; Sampling theory, sample surveys; Sciences and techniques of general use; Simulation; Simulations; Statistical methods; Statistical variance; Statistics; Stochastic processes; Studies; Survey sampling; Unbiased estimators
• ISSN: 0006-3444; 1464-3510
• DOI: 10.1093/biomet/asm098

We consider the problem of estimating the population total in two-stage cluster sampling when cluster sizes are known only for the sampled clusters, making use of a population model arising from a variance component model. The problem can be considered as one of predicting the unobserved part Z of the total, and the concept of predictive likelihood is studied. Prediction intervals and a predictor for the population total are derived for the normal case, based on predictive likelihood. For a more general distribution-free model, by application of an analysis of variance approach instead of maximum likelihood for parameter estimation, the predictor obtained from the predictive likelihood is shown to be approximately uniformly optimal for large sample size and large number of clusters, in the sense of uniformly minimizing the mean-squared error in a partially linear class of model-unbiased predictors. Three prediction intervals for Z based on three similar predictive likelihoods are studied. For a small number n0 of sampled clusters, they differ significantly, but for large n0, the three intervals are practically identical. Model-based and design-based coverage properties of the prediction intervals are studied based on a comprehensive simulation study. The simulation study indicates that for large sample sizes, the coverage measures achieve approximately the nominal level 1 − α and are slightly less than 1 − α for moderately large sample sizes. For small sample sizes, the coverage measures are about 1 − 2α, being raised to 1 − α for a modified interval based on the distribution.