On the structure of UMVUEs

TL;DR

The paper explores the structure of uniformly minimum variance unbiased estimators (UMVUEs), introducing MVE-algebras and comparing them to subalgebras generated by complete sufficient statistics, revealing new statistical structures.

Contribution

It characterizes the structure of UMVUEs through MVE-algebras and identifies cases where these differ from traditional subalgebras, unveiling new statistical insights.

Findings

Existence of a subalgebra al U such that all al U-measurable statistics with finite second moments are UMVUEs

MVE-algebras are similar to subalgebras generated by complete sufficient statistics

Examples where these subalgebras differ, leading to new statistical structures

Abstract

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.