Optimal local Gaussian approximation of an exponential family (Q1085905)

Under certain regularity conditions products \({\mathcal E}^ n\) of an experiment \({\mathcal E}\) can be locally approximated by homoscedastic Gaussian experiments \({\mathcal G}_ n\). \({\mathcal G}_ n\) can be defined such that the square roots of the densities have nearly the same structure with respect to the \(L^ 2\)-geometry as in \({\mathcal E}^ n\). The main result of this paper is that this choice of \({\mathcal G}_ n\) is asymptotically optimal in the sense of minimizing the deficiency distance between \({\mathcal E}^ n\) and \({\mathcal G}\) if \({\mathcal E}\) is a one-dimensional exponential family.