## Analytic Robust Inference for Small Area Means Using Student t Prior Distributions

### by Richard Evans.

**Abstract:**
Hierarchical models for L studies, domains or experiments often assume
that the study means have a common normal population distribution. However,
modeling normal sampling distributions with a normal population distribution may
overstate the level of exchangeability of the studies and cause "borrowing of
strength" among dissimilar domains. When there is uncertainty about the
similarity of the domains, using heavy tailed population distributions, in
particular t distributions, provide some protection from combining dissimilar
studies, domains or experiments (Gelman, Carlin, Stern, and Rubin, 1995,
O'Hagan, 1988). We also use t population distributions, but use an analytic
method instead of approximations or Monte Carlo methods to provide posterior
inference. We expand the current analytic inferential methodology to the
situation when the sampling variance and the prior scale are unknown. In the
case that the prior parameters are unknown the analytic method may be modified
to provide a parametric empirical Bayes method for inference about the study
means. The analytic method circumvents Markov chain Monte Carlo convergence
problems and permits a more direct model sensitivity analysis. Using examples
and analytic results we demonstrate the characteristics of posterior means and
variances under t distribution priors, and suggest that for applied data
analysis problems the Cauchy prior is a reasonable population distribution. The
examples also suggest that the prior variance should be estimated from the data
rather than assigned an arbitrary constant. Finally we demonstrate the method
using real data in a small area estimation example.
**Key Words: **
Heavy tails, empirical Bayes, complex analysis, robust inference,
small area estimation

**Author:**

Richard Evans, revans@iastate.edu

**Editor:**
Efthymios G. Tsionas,tsionas@aueb.gr

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