A characterization of best unbiased estimators

TL;DR

This paper provides a simple linear algebra-based characterization of uniformly minimum variance unbiased estimators (UMVUEs) for finite sample spaces, extending to estimators optimal under convex loss functions.

Contribution

It introduces a novel linear independence condition for likelihood functions that characterizes UMVUEs and extends this to estimators minimizing arbitrary convex loss functions.

Findings

Likelihood functions are eigenvectors of a constructed matrix.

UMVUE existence implies a specific eigenstructure.

Characterization applies to convex loss functions.

Abstract

A simple characterization of uniformly minimum variance unbiased estimators (UMVUEs) is provided (in the case when the sample space is finite) in terms of a linear independence condition on the likelihood functions corresponding to the possible samples. The crucial observation in the proof is that, if a UMVUE exists, then, after an appropriate cleaning of the parameter space, the nonzero likelihood functions are eigenvectors of an "artificial" matrix of Lagrange multipliers, and the values of the UMVUE are eigenvalues of that matrix. The characterization is then extended to best unbiased estimators with respect to arbitrary convex loss functions.