On the Non-Existence of Unbiased Estimators in Constrained Estimation Problems

TL;DR

This paper proves that unbiased estimators cannot exist in constrained estimation problems when the parameter set is compact and distributions are absolutely continuous, with additional conditions and examples illustrating these results.

Contribution

It establishes fundamental non-existence results for unbiased estimators under specific constrained conditions, extending understanding in statistical estimation theory.

Findings

Unbiased estimators do not exist for compact parameter sets with absolutely continuous distributions.

Weaker conditions for non-existence are also identified.

Examples demonstrate the practical relevance of these theoretical results.

Abstract

We address the problem of existence of unbiased constrained parameter estimators. We show that if the constrained set of parameters is compact and the hypothesized distributions are absolutely continuous with respect to one another, then there exists no unbiased estimator. Weaker conditions for the absence of unbiased constrained estimators are also specified. We provide several examples which demonstrate the utility of these conditions.