Generalized Estimation and Information

TL;DR

This paper extends the concept of generalized estimators to multi-dimensional parameters, introducing an extended information framework to evaluate their properties, and demonstrates Fisher information as an upper bound for estimator information.

Contribution

It generalizes the theory of estimators and information measures to multi-dimensional cases, offering new assessment methods beyond traditional point estimator analysis.

Findings

Fisher information bounds the information used by estimators.

The score function attains the Fisher information bound.

Extended information measures are proposed for generalized estimators.

Abstract

This paper extends the idea of a generalized estimator for a scalar parameter (Vos, 2022) to multi-dimensional parameters both with and without nuisance parameters. The title reflects the fact that generalized estimators provide more than simply another method to find point estimators, and that the methods to assess generalized estimators differ from those for point estimators. By generalized estimation we mean the use of generalized estimators together with an extended definition of information to assess their inferential properties. We show that Fisher information provides an upper bound for the information utilized by an estimator and that the score attains this bound.