Some aspects of minimum variance unbiased estimation in presence of ancillary statistics

We show that the fact that any uniformly minimum variance unbiased estimator (UMVUE) is uncorrelated with all unbiased estimators of zero implies certain conclusions of the Rao–Blackwell and Lehmann–Scheffe theorems. We also show that the conditional mean and variance of any UMVUE, given any ancillary statistic, must be independent of the ancillary. We discuss some implications of these results when a minimal sufficient statistic is incomplete and contains ancillaries.