Existence and uniqueness of weighted generalized ψ-estimators
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We introduce the notions of generalized and weighted generalized ψ-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 ψ-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 ψ-estimators that are important in robust statistics, and some examples from maximum likelihood theory as well. Further, we introduce Bajraktarević-type (in particular, quasi-arithmetic-type) ψ-estimators. Our results specialized to ψ-estimators with a function ψ being continuous in its second variable provide new results for (usual) ψ-estimators (also called Z-estimators).
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