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...

The aso theory developed in Chapters 8 - 12 presumes that the tan gent cones are linear spaces. In the present chapter we collect a few natural examples where the tangent cone fails to be a linear space. These examples are to remind the reader that an extension of the theo ry to convex tangent cones is wanted. Since the results are not needed in the rest of the book, we are more generous ab out regularity condi tions. The common feature of the examples is the following: Given a pre order (i.e., a reflexive and transitive order relation) on a family of p-measures, and a subfamily i of order...