Generalized estimation and information

Abstract This paper extends the idea of a generalized estimator for a scalar parameter (Vos in Inform Geom 7:151–170, 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. This optimality result provides theoretical justification for likelihood-based inference, effectively narrowing the search for optimal estimators when the score is computationally feasible.