Efficiency and Robustness in a Geometrical Perspective

A geometrical setting is constructed, based on Hilbert space, in which the asymptotic properties of estimators can be studied. Estimators are defined in the context of parametrised models, which are treated as submanifolds of an underlying Hilbert manifold, on which a parameter-defining mapping is defined as a submersion on to a finite-dimensional parameter space. Robustness of an estimator is defined as its root-$n$ consistency at all points in the model, and efficicency is based on the criterion, natural in the Hilbert space setting, of the asymptotic variance.