UMVUE-Type Estimators under Bregman Losses

2026-05-08

AI Summary

Existing theory lacks a unified optimal framework for unbiased estimation under Bregman losses. This work addresses this gap by leveraging the bias–variance decomposition induced by Bregman divergences to introduce two natural classes of loss functions and corresponding notions of unbiasedness. Focusing on the first class, the paper establishes a novel theoretical framework for Bregman-unbiased estimation by characterizing unbiasedness in the dual space and deriving analogues of the Rao–Blackwell and Lehmann–Scheffé theorems. By integrating tools from convex analysis, sufficient statistics, and classical unbiased estimation theory, the study systematically constructs the first nontrivial examples of uniformly minimum Bregman-variance unbiased estimators (UMVUEs) under this framework, thereby substantially extending the scope of classical UMVUE theory beyond squared-error loss.

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, $D_{\varphi}(θ,\hatθ)$ and $D_{\varphi}(\hatθ,θ)$, 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{é} theorems, providing a systematic construction of type-I Bregman UMVUEs.

Problem

• UMVUE
• Bregman loss
• unbiased estimation
• Bregman divergence
• dual space

Innovation

• Bregman loss
• UMVUE
• unbiased estimation
• dual space
• Rao–Blackwell theorem