The structure of the UMVUEs from categorical data

Let an observation $X$ take finitely many values with probabilities $p_1(\theta),\ldots,p_N(\theta)$ depending on an abstract parameter $\theta\in\Theta$. It is proved that a statistic is a uniformly minimum variance unbiased estimator (UMVUE) if and only if it is measurable with respect to a subalgebra of the finite algebra generated by $X$. In general, this subalgebra is smaller than the minimal sufficient subalgebra for $\theta$ and is explicitly described. It is related to a special partition of a finite set of elements of an abstract linear space.