some properties of posterior pitman closeness

This article presents some results on a Bayesian notion of Pitman Closeness, defined in Ghosh and Sen (1991) and called Posterior Pitman Closeness (PPC). Their criterion avoids some of the drawbacks of the well-known (frequentist) Pitman closeness criterion, as introduced by Pitman (1937). It is shown that, if two estimators have the same posterior distribution of the distance from θ, the posterior distribution of θ has to be symmetric. This implies, in particular, that the estimators are Posterior Pitman equivalent. It is also shown that the PPC criterion does not suffer from another paradoxical property illustrated by Blyth and Pathak (1985) - that of an estimator δ₁ being stochastically closer to a parameter θ than another estimator δ₂ and yet being Pitman closer to θ than δ₁. It turns out that, if δ₁ is stochastically closer to θ than δ₂, conditional on x, then it is also Posterior Pitman closer.

We show that the original multivariate concept of PPC is no longer transitive. We provide necessary and sufficient conditions for a Posterior Pitman closest estimator to exist, thus generalizing Theorems 2 and 3 of Ghosh and Sen (1991). We show that a Posterior Pitman closest estimator does not always exist in several dimensions.