UMVUE-Type Estimators under Bregman Losses
For statisticians and machine learning researchers, this work generalizes a fundamental estimation framework to a broader class of loss functions, but the results are theoretical and incremental relative to existing UMVUE theory.
The paper extends the theory of uniformly minimum variance unbiased estimators (UMVUEs) to Bregman losses, establishing analogs of the Rao-Blackwell and Lehmann-Scheffé theorems for a new notion of unbiasedness in the dual space. This provides a systematic construction of optimal estimators under Bregman divergences.
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, $D_φ(θ,\hatθ)$ and $D_φ(\hatθ,θ)$, 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 $\nablaφ$. 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.