Beyond Brown's radial tests: admissibility, parabolicity, and a doubling-Poincaré prior that neither test decides

Abstract

For the estimation of a normal mean in $\mathbb{R}^m$ under quadratic loss, Brown (1971) proved that a generalized Bayes estimator $δ_F$ is admissible if and only if a diffusion attached to the marginal $f^*$ of the prior $F$ is recurrent, and gave two radial tests: a sphere-average test for recurrence and a ray test for transience. He observed that both tests can be silent. We read his theory as the potential theory of the weighted Euclidean space $(\mathbb{R}^m,f^*\,dx)$ and ask what the modern theory of recurrence on weighted manifolds adds inside the zone where neither test applies. Three results. (A) If $f^*\,dx$ is doubling and supports a scale-invariant Poincaré inequality, the sphere test is already complete: recurrence, Brown's sphere test and the ball-volume test are equivalent, so geometric criteria can add decidable cases only on the inadmissible side. (B) They do add one. An explicit prior on $\mathbb{R}^3$ whose marginal is small along a lattice of circles at dyadic scales with spacing equal to the square root of the scale, is doubling and Poincaré, silences both of Brown's tests, and is transient by the ball-volume test; its generalized Bayes estimator is inadmissible. (C) Outside the doubling-Poincaré class the zone is two-sided: the same circles placed on one surface per scale give an admissible estimator, decided by a flux bound along a non-spherical exhaustion, and a logarithmic product prior gives an inadmissible one, decided by product heat kernels. A comparison lemma shows that any positive weight with Lipschitz logarithm is, up to constants, the marginal of a prior, so that explicit weights may be used throughout.