  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 solving likelihood equations in case of normal, a Beta-type,
Gamma, Lomax (Pareto type II), lognormal and Laplace distributions.
