Comparison and equality of generalized \(\psi \)-estimators

We solve the comparison problem for generalized \(\psi \) $\psi$ -estimators introduced by Barczy and Páles (arXiv: 2211.06026, 2022). Namely, we derive several necessary and sufficient conditions under which a generalized \(\psi \) $\psi$ -estimator less than or equal to another \(\psi \) $\psi$ -estimator for any sample. We also solve the corresponding equality problem for generalized \(\psi \) $\psi$ -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.