UMVUE-Type Estimators under Bregman Losses

TL;DR

This paper extends classical UMVUE theory to Bregman losses, introducing dual unbiasedness concepts and systematic construction methods for Bregman UMVUEs.

Contribution

It develops a new framework for unbiased estimation under Bregman divergences, including dual unbiasedness and Rao-Blackwell type theorems.

Findings

Established dual notions of unbiasedness in Bregman divergence settings.

Derived Rao-Blackwell and Lehmann–Scheffé type theorems for Bregman UMVUEs.

Provided a systematic method to construct Bregman UMVUEs.

Abstract

We study unbiased estimation under Bregman losses and develop an extension of the classical theory of uniformly minimum variance unbiased estimators (UMVUEs). Exploiting bias--variance-type decompositions for Bregman divergences, we consider two natural loss functions, $D_{\varphi}(\theta,\hat{\theta})$ and $D_{\varphi}(\hat{\theta},\theta)$, and their corresponding notions of unbiasedness. We show that the latter formulation reduces to the classical setting, whereas the former yields a different framework in which unbiasedness is characterized in the dual space induced by $\nabla\varphi$. For the nontrivial case, we establish analogs of the Rao--Blackwell and Lehmann--Scheff{\'e} theorems, providing a systematic construction of type-I Bregman UMVUEs.