  Small area estimation has become an important tool in official statistics,
used to construct estimates of population quantities for domains with small
sample sizes. Typical area-level models function as a type of heteroscedastic
regression, where the variance for each domain is assumed to be known and
plugged in following a design-based estimate. Recent work has considered
hierarchical models for the variance, where the design-based estimates are used
as an additional data point to model the latent true variance in each domain.
These hierarchical models may incorporate covariate information, but can be
difficult to sample from in high-dimensional settings. Utilizing recent
distribution theory, we explore a class of Bayesian hierarchical models for
small area estimation that smooth both the design-based estimate of the mean
and the variance. In addition, we develop a class of unit-level models for
heteroscedastic Gaussian response data. Importantly, we incorporate both
covariate information as well as spatial dependence, while retaining a
conjugate model structure that allows for efficient sampling. We illustrate our
methodology through an empirical simulation study as well as an application
using data from the American Community Survey.
