  In all setups when the structure of UMVUEs is known, there exists a
subalgebra $\cal U$ (MVE-algebra) of the basic $\sigma$-algebra such that all
$\cal U$-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.
