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The two-level normal hierarchical model has played an important role instatistical theory and applications. In this paper, we first introduce ageneral adjusted maximum likelihood method for estimating the unknown variancecomponent of the model and the associated empirical best linear unbiasedpredictor of the random effects. We then discuss a new idea for selecting priorfor the hyperparameters. The prior, called a multi-goal prior, producesBayesian solutions for hyperparmeters and random effects that match (in thehigher order asymptotic sense) the corresponding classical solution in linearmixed model with respect to several properties. Moreover, we establish for thefirst time an analytical equivalence of the posterior variances under theproposed multi-goal prior and the corresponding parametric bootstrapsecond-order mean squared error estimates in the context of a random effectsmodel.

Multi-Goal Prior Selection: A Way to Reconcile Bayesian and Classical Approaches for Random Effects Models