Perturbation Invariant Estimates and Incidental Nuisance Parameters

It is shown (Proposition (3.9)) that the asymptotic information bound which is valid for the estimation of a parameter in the structure (mixture) model remains valid in the functional model (incidental nuisance parameters) if only perturbation symmetric estimators (Deenition (3.6)) are admitted. Perturbation symmetry is a property which is closely related to permutation symmetry (Theorem (3.4)). In particular, equicontinuous functions of empirical processes are perturbation symmetric (Theorem (3.3)). Thus, the results of this paper continue a discussion initiated by Bickel and Klaassen, 2], Pfanzagl, 14], and Strasser, 21], on permutation symmetry of estimators and the exclusion of supereeciency in the functional model.