This paper considers parametric estimation problems
with independent, identically nonregularly
distributed data. It focuses 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. These rates are shown to
be attainable in general classes of parametric
estimation problems.