Robustness of the posterior mean in normal hierarchical models

We consider the problem of robustness in hierarchical Bayes models. Let X = (X₁,X₂, … ,X_p)^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.