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 (Definition (3.6)) are admitted. Pertur- bation 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 (1986), Pfanzagl (1993) and Strasser (1996) on permutation symmetry of estimators and the exclusion of superefficiency in the functional model. (authors' abstract)