Robustness of the posterior mean in normal hierarchical models

We consider the problem of robustness in hierarchical Bayes models. Let X = (X1,X2, … ,Xp)τ be a random vector, the X1 being independently distributed as N(θ1,σ2) random variables (σ2 known), while the θ1 are thought to be exchangeable, modelled as i.i.d, N(μ,τ2). The hyperparameter µ is given a noninformative prior distribution π(μ) = 1 and τ2 is assumed to be independent of µ having a distribution g(τ2) lying in a certain class of distributions g. For several g's, including e-contaminations classes and density ratio classes we determine the range of the posterior mean of θ1 as g ranges over g.

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