On perturbations of strongly admissible prior distributions

Consider a parametric statistical model, P (d x | θ), and an improper prior distribution, ν (d θ), that together yield a (proper) formal posterior distribution, Q (d θ | x). The prior is called strongly admissible if the generalized Bayes estimator of every bounded function of θ is admissible under squared error loss. Eaton [M.L. Eaton, A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains, Annals of Statistics 20 (1992) 1147-1179] used the Blyth-Stein Lemma to develop a sufficient condition, call it C, for strong admissibility of ν. Our main result says that, under mild regularity conditions, if ν satisfies C and g (θ) is a bounded, non-negative function, then the perturbed prior distributiong (θ) ν (d θ) also satisfies C and is therefore strongly admissible. Our proof has three basic components: (i) Eaton's [M.L. Eaton, A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains, Annals of Statistics 20 (1992) 1147-1179] result that the condition C is equivalent to the local recurrence of the Markov chain whose transition function is R (d θ | η) = ∫ Q (d θ | x) P (d x | η); (ii) a new result for general state space Markov chains giving conditions under which local recurrence is equivalent to recurrence; and (iii) a new generalization of Hobert and Robert's [J.P. Hobert, C.P. Robert, Eaton's Markov chain, its conjugate partner and P-admissibility, Annals of Statistics 27 (1999) 361-373] result that says Eaton's Markov chain is recurrent if and only if the chain with transition function over(R, ̃) (d x | y) = ∫ P (d x | θ) Q (d θ | y) is recurrent. One important application of our results involves the construction of strongly admissible prior distributions for estimation problems with restricted parameter spaces.