Statistica Sinica 9(1999), 325-343

AN EMPIRICAL BAYES PREDICTION INTERVAL FOR THE

FINITE POPULATION MEAN OF A SMALL AREA

Balgobin Nandram

Worcester Polytechnic Institute

Abstract: We construct an empirical Bayes (EB) prediction interval for the finite population mean of a small area when data are available from many similar small areas. We assume that the individuals of the population of the ith area are a random sample from a normal distribution with mean μi and variance σi2. Then, given σi2, the μi are independently distributed with each μi having a normal distribution with mean θ and variance σi2τ , and the σi2 are a random sample from an inverse gamma distribution with index η and scale (η-1)δ . First, assuming θ, τ, δ and η are fixed and known, we obtain the highest posterior density (HPD) interval for the finite population mean of the lth area. Second, we obtain the EB interval by ``substituting'' point estimators for the fixed and unknown parameters θ, τ, δ and η into the HPD interval, and a two-stage procedure is used to partially account for underestimation of variability. Asymptotic properties (as l → ∞) of the EB interval are obtained by comparing its center, width and coverage probability with those of HPD interval. Finally, by using a small-scale numerical study, we assess the asymptotic properties of the proposed EB interval, and we show that the EB interval is a good approximation to the HPD interval for moderate values of l.

Key words and phrases: Asymptotic, Bayes risk, Monte Carlo, HPD interval, simulation, uniform integrability.