On perturbations of strongly admissible prior distributions

Abstract

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 distribution g ( 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 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.