Impact of Existence and Nonexistence of Pivot on the Coverage of Empirical Best Linear Prediction Intervals for Small Areas

We advance parametric bootstrap theory to construct highly efficient empirical best prediction intervals for small area means, achieving a coverage error rate of O(m−3/2), where <i>m</i> is the number of areas modeled by a linear mixed normal model. For a general mixed effect model with random effects from a known but nonnormal distribution (with unknown hyperparameters), we show analytically that the empirical best linear (EBL) prediction interval maintains the same coverage error order, assuming the existence of a pivot for standardized random effects with known hyperparameters. Recognizing challenges in proving pivot existence, we develop a simple moment-based method to claim nonexistence of pivot. We find that without a pivot, the parametric bootstrap EBL prediction interval fails to reach the desired O(m−3/2) coverage error. We obtain a surprising result that the order O(m−1) term is always positive under certain conditions indicating possible overcoverage of the existing parametric bootstrap EBL prediction interval. In general, we analytically show for the first time that the coverage problem can be corrected by adopting a suitably devised double parametric bootstrap. Our Monte Carlo simulations show that our proposed single bootstrap method performs reasonably well when compared to rival methods. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.