Assume one has to estimate the mean J x P( dx) (or the median of P, or
any other functional t;;(P)) on the basis ofi.i.d. observations from P.
Ifnothing is known about P, then the sample mean is certainly the best
estimator one can think of. If P is known to be the member of a certain
parametric family, say {Po: {) E e}, one can usually do better by
estimating {) first, say by {)(n)(. .), and using J XPo(n)(;r.) (dx) as
an estimate for J xPo(dx). There is an "intermediate" range, where we
know something about the unknown probability measure P, but less than
parametric theory takes for granted. Practical problems have always led
statisticians to invent estimators for such intermediate models, but it
usually remained open whether these estimators are nearly optimal or
not. There was one exception: The case of "adaptivity", where a
"nonparametric" estimate exists which is asymptotically optimal for any
parametric submodel. The standard (and for a long time only) example of
such a fortunate situation was the estimation of the center of symmetry
for a distribution of unknown shape.