<p>We introduce the notions of generalized and weighted generalized <i>ψ</i>-estimators as unique points of sign change of some appropriate functions and give necessary and sufficient conditions for their existence. We also derive a set of sufficient conditions under which the so-called <i>ψ</i>-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 <i>ψ</i>-estimators that are important in robust statistics, and some examples from maximum likelihood theory. Further, we introduce Bajraktarević-type (in particular, quasiarithmetic-type) <i>ψ</i>-estimators. Our results specialized to <i>ψ</i>-estimators with a function <i>ψ</i> continuous in its second variable provide new results for (usual) <i>ψ</i>-estimators (also called Z-estimators).</p>