Existence and uniqueness of weighted generalized $\psi$-estimators

We introduce the notions of generalized and weighted generalized $\psi$-estimators as unique points of sign change of some appropriate functions, and we give necessary as well as sufficient conditions for their existence. We also derive a set of sufficient conditions under which the so-called $\psi$-expectation function has a unique point of sign change. We present several examples from statistical estimation theory, where our results are well-applicable. For example, we consider the cases of empirical quantiles, empirical expectiles, some $\psi$-estimators that are important in robust statistics, and some examples from maximum likelihood theory as well. Further, we introduce Bajraktarevi\'c-type (in particular, quasi-arithmetic-type) $\psi$-estimators. Our results specialized to $\psi$-estimators with a function $\psi$ being continuous in its second variable provide new results for (usual) $\psi$-estimators (also called Z-estimators).