Introduction

Abstract

Assume one has to estimate the mean ∫ x P(dx) (or the median of P, or any other functional К(P)) on the basis of i.i.d. observations from P. If nothing 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 {P_ϑ}: ϑ ∈ Θ, one can usually do better by estimating ϑ first, say by ϑ⁽ⁿ⁾(x), and using ∫ x P _{ϑ (n)}(x) (dx) as an estimate for ∫ x P ϑ(dx). There is an “intermediate” range, where we know something about the unknown probability measure P, but less than parametric theory takes for granted.