Inference for a constrained parameter in presence of an uncertain constraint

TL;DR

This paper develops a hierarchical Bayesian method for inferring a parameter constrained by an uncertain lower bound, providing estimators that are minimax under squared error loss in normal and Poisson models.

Contribution

It introduces a novel Bayesian framework for parameters with uncertain constraints, deriving identities and estimators that respect the constraints while maintaining optimality.

Findings

Bayes estimators are minimax under squared error loss.

Method applies to normal and Poisson models.

Estimators adapt to uncertain lower bounds.

Abstract

We describe a hierarchical Bayesian approach for inference about a parameter $\theta$ lower-bounded by $\alpha$ with uncertain $\alpha$, derive some basic identities for posterior analysis about $(\theta,\alpha)$, and provide illustrations for normal and Poisson models. For the normal case with unknown mean $\theta$ and known variance $\sigma^2$, we obtain Bayes estimators of $\theta$ that take values on $\mathbb{R}$, but that are equally adapted to a lower-bound constraint in being minimax under squared error loss for the constrained problem.