  We introduce a general hierarchical Bayesian framework that incorporates a
flexible nonparametric data model specification through the use of empirical
likelihood methodology, which we term semiparametric hierarchical empirical
likelihood (SHEL) models. Although general dependence structures can be readily
accommodated, we focus on spatial modeling, a relatively underdeveloped area in
the empirical likelihood literature. Importantly, the models we develop
naturally accommodate spatial association on irregular lattices and irregularly
spaced point-referenced data. We illustrate our proposed framework by means of
a simulation study and through three real data examples. First, we develop a
spatial Fay-Herriot model in the SHEL framework and apply it to the problem of
small area estimation in the American Community Survey. Next, we illustrate the
SHEL model in the context of areal data (on an irregular lattice) through the
North Carolina sudden infant death syndrome (SIDS) dataset. Finally, we analyze
a point-referenced dataset from the North American Breeding Bird survey that
considers dove counts for the state of Missouri. In all cases, we demonstrate
superior performance of our model, in terms of mean squared prediction error,
over standard parametric analyses.
