Fisher Information in Group-Type Models

TL;DR

This paper demonstrates that in smooth parametric group models, the existence of finite Fisher information implies the densities are in fact consequences of differentiability conditions, linking Fisher information to local asymptotic normality.

Contribution

It generalizes Huber's theorem to show that finite Fisher information in group models ensures the existence of densities and L_2-differentiability, with applications to location-scale models.

Findings

Finiteness of Fisher information implies L_2-differentiability.

In location-scale models, Fisher information finiteness is equivalent to local asymptotic normality.

Existence and uniqueness of Fisher information minimizers are established under certain conditions.

Abstract

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.