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

Richard Evans,

Editor: Efthymios G. Tsionas,

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