On the Hierarchical Bayes justification of Empirical Bayes Confidence Intervals

Multi-level normal hierarchical models, also interpreted as mixed effects models, play an important role in developing statistical theory in multi-parameter estimation for a wide range of applications. In this article, we propose a novel reconciliation framework of the empirical Bayes (EB) and hierarchical Bayes approaches for interval estimation of random effects under a two-level normal model. Our framework shows that a second-order efficient empirical Bayes confidence interval, with EB coverage error of order $O(m^{-3/2})$, $m$ being the number of areas in the area-level model, can also be viewed as a credible interval whose posterior coverage is close to the nominal level, provided a carefully chosen prior - referred to as a 'matching prior' - is placed on the hyperparameters. While existing literature has examined matching priors that reconcile frequentist and Bayesian inference in various settings, this paper is the first to study matching priors with the goal of interval estimation of random effects in a two-level model. We obtain an area-dependent matching prior on the variance component that achieves a proper posterior under mild regularity conditions. The theoretical results in the paper are corroborated through a Monte Carlo simulation study and a real data analysis.