Inference for a constrained parameter in presence of an uncertain constraint
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We describe a hierarchical Bayesian approach for inference about a parameter θ lower-bounded by α with uncertain α, derive some basic identities for posterior analysis about (θ,α), and provide illustrations for normal and Poisson models. For the normal case with unknown mean θ and known variance σ^2, we obtain Bayes estimators of θ that take values on R, but that are equally adapted to a lower-bound constraint in being minimax under squared error loss for the constrained problem.
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