  The two-level normal hierarchical model has played an important role in
statistical theory and applications. In this paper, we first introduce a
general adjusted maximum likelihood method for estimating the unknown variance
component of the model and the associated empirical best linear unbiased
predictor of the random effects. We then discuss a new idea for selecting prior
for the hyperparameters. The prior, called a multi-goal prior, produces
Bayesian solutions for hyperparmeters and random effects that match (in the
higher order asymptotic sense) the corresponding classical solution in linear
mixed model with respect to several properties. Moreover, we establish for the
first time an analytical equivalence of the posterior variances under the
proposed multi-goal prior and the corresponding parametric bootstrap
second-order mean squared error estimates in the context of a random effects
model.
