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We employ a parameter-free distribution estimation framework where estimatorsare random distributions and utilize the Kullback-Leibler (KL) divergence as aloss function. Wu and Vos [J. Statist. Plann. Inference 142 (2012) 1525-1536]show that when an estimator obtained from an i.i.d. sample is viewed as arandom distribution, the KL risk of the estimator decomposes in a fashionparallel to the mean squared error decomposition when the estimator is areal-valued random variable. In this paper, we explore how conditional versionsof distribution expectation ($E^{\dagger}$) can be defined so that adistribution version of the Rao-Blackwell theorem holds. We definedistributional expectation and variance ($V^{\dagger}$) that also provide adecomposition of KL risk in exponential and mixture families. For exponentialfamilies, we show that the maximum likelihood estimator (viewed as a randomdistribution) is distribution unbiased and is the unique uniformly minimumdistribution variance unbiased (UMV$^{\dagger}$U) estimator. Furthermore, weshow that the MLE is robust against model specification in that if the truedistribution does not belong to the exponential family, the MLE isUMV$^{\dagger}$U for the KL projection of the true distribution onto theexponential families provided these two distribution have the same expectationfor the canonical statistic. To allow for estimators taking values outside ofthe exponential family, we include results for KL projection and define anextended projection to accommodate the non-existence of the MLE for familieshaving discrete sample space. Illustrative examples are provided.

Maximum likelihood estimators uniformly minimize distribution variance among distribution unbiased estimators in exponential families