Empirical Bayes Estimation for the Finite Population Mean on the Current Occasion

Abstract

Many finite populations which are sampled repeatedly change slowly over time. Then estimation of finite population characteristics for the current occasion, l, may be improved by the use of data from previous surveys. In this article we investigate the use of empirical Bayes procedures based on two superpopulation models. Each model has the same first stage: The values of the population units on the ith occasion are a random sample from the normal distribution with mean μ _i and variance σ ² _i . At the second stage we assume that either (a) μ ₁, …, μ _l are a random sample from the normal distribution with mean θ and variance δ ², or (b) given σ ² _i and τ, μ _i has the normal distribution with mean θ and variance σ ² _i τ (independently for each i), whereas the σ ² _i are a random sample from the inverse gamma distribution with parameters η/2 and κ/2. In (a) the σ ² _i , θ, and σ ² are assumed to be unknown, whereas in (b) θ, τ, and κ are unknown. We develop empirical Bayes point estimators and confidence intervals for the finite population mean on the lth occasion and make large-sample comparisons with the corresponding Bayes estimators and intervals. These are asymptotic results obtained within the framework of “classical” empirical Bayes theory. To complement the asymptotic results we present the results of an extensive numerical investigation of the properties of these estimators and intervals when sample sizes are moderate. The methodology described here is also appropriate for “small area” estimation.

Key Words:

• Analysis of variance estimators
• Confidence interval
• Simulation
• Small area estimation