Asymptotic admissibility of priors and elliptic differential equations

TL;DR

This paper links the asymptotic behavior of estimators to elliptic differential equations, establishing criteria for the admissibility of priors based on boundary conditions and providing explicit examples.

Contribution

It introduces a novel connection between asymptotic estimator risk analysis and elliptic differential equations, characterizing prior admissibility in a rigorous mathematical framework.

Findings

Admissibility depends on solutions to elliptic differential equations.

The unique admissible invariant prior for V=I, D=R^d-{0} is identified.

A detailed example with a normal mixture model illustrates the theory.

Abstract

We evaluate priors by the second order asymptotic behavior of the corresponding estimators.Under certain regularity conditions, the risk differences between efficient estimators of parameters taking values in a domain D, an open connected subset of R^d, are asymptotically expressed as elliptic differential forms depending on the asymptotic covariance matrix V. Each efficient estimator has the same asymptotic risk as a 'local Bayes' estimate corresponding to a prior density p. The asymptotic decision theory of the estimators identifies the smooth prior densities as admissible or inadmissible, according to the existence of solutions to certain elliptic differential equations. The prior p is admissible if the quantity pV is sufficiently small near the boundary of D. We exhibit the unique admissible invariant prior for V=I,D=R^d-{0). A detailed example is given for a normal mixture model.