Pfanzagl J. Estimation in Semiparametric Models: Some Recent Developments

New York: Springer, 1990. — 115 p. — (Lecture Notes in Statistics 63). — ISBN: 978-1-4612-3396-1.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.Tangent spaces and gradients
Asymptotic bounds for the concentration of estimator-sequences
Constructing estimator-sequences
Estimation in semiparametric models
Families of gradients
Estimating equations
A special semiparametric model
Mixture models
Examples of mixture models
Auxiliary results