Maximal Uniform Convergence Rates in Parametric Estimation Problems

This paper considers parametric estimation problems with i.i.d. data. It focusses on rate-efficiency, in the sense of maximal possible convergence rates of stochastically bounded estimators, as an optimality criterion, largely unexplored in parametric estimation. Under mild conditions, the Hellinger metric, defined on the space of parametric probability measures, is shown to be an essentially universally applicable tool to determine maximal possible convergence rates.