Comparison and equality of generalized $\psi$-estimators

TL;DR

This paper establishes conditions for comparing and equating generalized $$-estimators, with applications to various statistical estimators and distributions, enhancing understanding of their relative performance and properties.

Contribution

It provides necessary and sufficient conditions for the comparison and equality of generalized $$-estimators, extending to specific estimator types and distributions.

Findings

Derived conditions for estimator comparison and equality.

Applied results to empirical expectiles and likelihood-based estimators.

Analyzed estimators across multiple distributions like normal, Beta, and Gamma.

Abstract

We solve the comparison problem for generalized $\psi$-estimators introduced in Barczy and P\'ales (2022). Namely, we derive several necessary and sufficient conditions under which a generalized $\psi$-estimator less than or equal to another $\psi$-estimator for any sample. We also solve the corresponding equality problem for generalized $\psi$-estimators. For applications, we solve the two problems in question for Bajraktarevi\'c-type- and quasi-arithmetic-type estimators. We also apply our results for some known statistical estimators such as for empirical expectiles and Mathieu-type estimators and for solutions of likelihood equations in case of normal, a Beta-type, Gamma, Lomax (Pareto type II), lognormal and Laplace distributions.