On the structure of UMVUEs

In all setups when the structure of UMVUEs is known, there exists a subalgebra $\mathcal{U}$ (MVE-algebra) of the basic $\sigma$-algebra such that all $\mathcal{U}$-measurable statistics with finite second moments are UMVUEs. It is shown that MVE-algebras are, in a sense, similar to the subalgebras generated by complete sufficient statistics. Examples are given when these subalgebras differ, in these cases a new statistical structure arises.