Fisher Information in Group-Type Models

In proofs of L_2-differentiability, Lebesgue densities of a central distribution are often assumed right from the beginning. Generalizing Theorem 4.2 of Huber[81], we show that in the class of smooth parametric group models these densities are in fact consequences of a finite Fisher information of the model, provided a suitable representation of the latter is used. The proof uses the notions of absolute continuity in k dimensions and weak differentiability. As examples to which this theorem applies, we spell out a number of models including a correlation model and the general multivariate location and scale model. As a consequence of this approach, we show that in the (multivariate) location scale model, finiteness of Fisher information as defined here is in fact equivalent to L_2-differentiability and to a log-likelihood expansion giving local asymptotic normality of the model. Paralleling Huber's proofs for existence and uniqueness of a minimizer of Fisher information to our situation, we get existence of a minimizer in any weakly closed set of central distributions F. If, additionally to analogue assumptions to those of Huber[81], a certain identifiability condition for the transformation holds, we obtain uniqueness of the minimizer. This identifiability condition is satisfied in the multivariate location scale model.