Bootstrap adjustments for empirical Bayes interval estimates of small-area proportions

Empirical Bayes approaches have often been applied to the problem of estimating small-area parameters. As a compromise between synthetic and direct survey estimators, an estimator based on an empirical Bayes procedure is not subject to the large bias that is sometimes associated with a synthetic estimator, nor is it as variable as a direct survey estimator. Although the point estimates perform very well, naive empirical Bayes confidence intervals tend to be too short to attain the desired coverage probability, since they fail to incorporate the uncertainty which results from having to estimate the prior distribution. Several alternative methodologies for interval estimation which correct for the deficiencies associated with the naive approach have been suggested. Laird and Louis (1987) proposed three types of bootstrap for correcting naive empirical Bayes confidence intervals. Calling the methodology of Laird and Louis (1987) an unconditional bias-corrected naive approach, Carlin and Gelfand (1991) suggested a modification to the Type In parametric bootstrap which corrects for bias in the naive intervals by conditioning on the data. Here we empirically evaluate the Type TI and Type III bootstrap proposed by Laird and Louis, as well as the modification suggested by Carlin and Gelfand (1991), with the objective of examining coverage properties of empirical Bayes confidence intervals for small-area proportions.