Axiomatic characterisation of generalized $ψ$-estimators

We give axiomatic characterisations of generalized $ψ$-estimators and (usual) $ψ$-estimators (also called $Z$-estimators), respectively. The key properties of estimators that come into play in the characterisation theorems are the symmetry, the (strong) internality and the asymptotic idempotency. In the proofs, a separation theorem for Abelian subsemigroups plays a crucial role.

💡 Research Summary

The paper addresses a foundational problem in the theory of M‑estimators, namely the axiomatic characterisation of generalized ψ‑estimators (also called Z‑estimators when the estimating equation is satisfied exactly). Starting from the classical definition introduced by Huber—an estimator ϑₙ,ψ is defined as a solution t∈Θ of the equation ∑_{i=1}^n ψ(ξ_i,t)=0—the authors previously studied existence and uniqueness of such estimators under mild sign‑change conditions on ψ. The remaining open question is whether an arbitrary estimator can be represented as a ψ‑estimator for some ψ.

To answer this, the authors introduce a collection of functional properties for ψ: continuity in the second argument (