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.
|Original language||English (US)|
|Number of pages||21|
|Journal||Annales de l'institut Henri Poincare (B) Probability and Statistics|
|State||Published - Sep 2007|
Bibliographical noteFunding Information:
The authors are grateful to an anonymous referee for helpful comments and suggestions and to Christian Robert for the French translation of the abstract. Hobert’s research partially supported by NSF Grants DMS-00-72827 and DMS-05-03648.
- Dirichlet form
- Formal Bayes rule
- Formal posterior distribution
- Improper prior distribution
- Restricted parameter space
- Symmetric Markov chain