I have as corollary to the Rao-Blackwell theorem: If a minimum variance unbiased estimator $\hat \theta$ for $\theta$ exists, there is a function $\hat \theta_T$ of the minimal sufficient statistic $T$ for $\theta$ which is a minimum variance unbiased estimator.
Do we need $T$ to be minimal sufficient though? It seems to me that just being sufficient is enough, but whenever my lecturer used this result in practice he showed that the $T$ wasn't just sufficient, but minimal sufficient.
