Estimating a Finite Population Mean from Unequal Probability Samples

Abstract

Two approaches to estimating a finite population mean from unequal probability samples are contrasted. In the prediction approach a model is specified for the population values and is used to predict the nonsampled values. In the generalized regression approach the Horvitz-Thompson estimator of the population mean is modified to allow the introduction of covariate information (Särndal 1980). Generalized regression estimates have the desirable property of asymptotic design consistency, which is not always enjoyed by estimates in the prediction class. However, it is suggested that the prediction approach is the more principled method of estimation. If asymptotic design consistency is a desirable property in a given application, then only models that yield asymptotically design-consistent prediction estimates should be used. Two classes of models with this property are suggested, namely fixed and random-effects models that allow a separate intercept for subclasses of the population indexed by the probability of selection. Estimates based on a simple random-effects model perform well in a limited simulation study carried out to illustrate some of the compared estimators.

Key Words:

• Unequal probability sampling
• Prediction models
• Finite population
• Random effects model
• Small area estimation
• Horvitz-Thompson estimator