Approximation and information topologies

In this paper approximation properties of finite dimensional parametric models are described in terms of an information metric: the Hellinger distance. Under conditions on the parametric family given solely in terms of a comparison of the Hellinger distance with the parameter metric, optimal rates of convergence are described. It is also shown how to use these conditions on the parametric family to determine whether consistent estimation is possible. We give applications of the theorems to regular and non-regular parametric families, and to nonlinear regression.