An empirical Bayes prediction interval for the finite population mean of a small area

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 i(th) area are a random sample from a normal distribution with mean mu(i) and variance sigma(i)(2). Then, given sigma(i)(2), the mu(i) are independently distributed with each mu(i) having a normal distribution with mean theta and variance sigma(i)(2)tau, and the sigma(i)(2) are a random sample from an inverse gamma distribution with index eta and scale (eta - 1)delta. First, assuming theta, tau, delta and eta 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 theta, tau, delta and eta into the HPD interval, and a two-stage procedure is used to partially account for underestimation of variability. Asymptotic properties (as l --> infinity) 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.