  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.
