A second-order efficient empirical Bayes confidence interval

TL;DR

This paper proposes a new second-order correct empirical Bayes confidence interval for normal means, which is shorter, computationally efficient, and more accurate than existing methods, demonstrated through simulation.

Contribution

It introduces a novel adjusted REML method for empirical Bayes confidence intervals that are second-order correct, shorter, and less computationally intensive than existing approaches.

Findings

The proposed interval has coverage error of order O(m^{-3/2})

It is always shorter than the direct confidence interval

Simulation shows it outperforms competing methods

Abstract

We introduce a new adjusted residual maximum likelihood method (REML) in the context of producing an empirical Bayes (EB) confidence interval for a normal mean, a problem of great interest in different small area applications. Like other rival empirical Bayes confidence intervals such as the well-known parametric bootstrap empirical Bayes method, the proposed interval is second-order correct, that is, the proposed interval has a coverage error of order $O(m^{-{3}/{2}})$. Moreover, the proposed interval is carefully constructed so that it always produces an interval shorter than the corresponding direct confidence interval, a property not analytically proved for other competing methods that have the same coverage error of order $O(m^{-{3}/{2}})$. The proposed method is not simulation-based and requires only a fraction of computing time needed for the corresponding parametric bootstrap empirical Bayes confidence interval. A Monte Carlo simulation study demonstrates the superiority of the proposed method over other competing methods.