UMVUE-Type Estimators under Bregman Losses

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, $D_{\ddot{I}}†(\hat{I}¸,\hat{\hat{I}}¸)$ and $D_{\ddot{I}}†(\hat{\hat{I}}¸,\hat{I}¸)$, 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 $\nabla \ddot{I}†$. 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.