  Random effects model can account for the lack of fitting a regression model
and increase precision of estimating area-level means. However, in case that
the synthetic mean provides accurate estimates, the prior distribution may
inflate an estimation error. Thus it is desirable to consider the uncertain
prior distribution, which is expressed as the mixture of a one-point
distribution and a proper prior distribution. In this paper, we develop an
empirical Bayes approach for estimating area-level means, using the uncertain
prior distribution in the context of a natural exponential family, which we
call the empirical uncertain Bayes (EUB) method. The regression model
considered in this paper includes the Poisson-gamma and the binomial-beta, and
the normal-normal (Fay-Herriot) model, which are typically used in small area
estimation. We obtain the estimators of hyperparameters based on the marginal
likelihood by using a well-known Expectation-Maximization algorithm, and
propose the EUB estimators of area means. For risk evaluation of the EUB
estimator, we derive a second-order unbiased estimator of a conditional mean
squared error by using some techniques of numerical calculation. Through
simulation studies and real data applications, we evaluate a performance of the
EUB estimator and compare it with the usual empirical Bayes estimator.
