Unbiasedness and Bayes Estimation

TL;DR

This paper explores the mathematical relationship between unbiased estimators and Bayes estimators under squared error loss, using the framework of linear operators in Hilbert spaces to reveal their connection.

Contribution

It introduces a novel operator-theoretic framework that unifies the concepts of unbiasedness and Bayesian estimation in a Hilbert space setting.

Findings

Unbiased and Bayes estimators can be represented as adjoint linear operators.

The operator framework provides new insights into estimator properties.

Implications for estimator design and analysis in statistical inference.

Abstract

Assuming squared error loss, we show that finding unbiased estimators and Bayes estimators can be treated as using a pair of linear operators that operate between two Hilbert spaces. We note that these integral operators are adjoint and then investigate some consequences of this fact.