UMVUE-Type Estimators under Bregman Losses

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, $D_{\phi }(\theta ,\widehat{\theta })$ and $D_{\phi }(\widehat{\theta },\theta )$, 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 $\mathrm{\nabla }\phi$. 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.