Estimation in Semiparametric Models

1. Front Matter

   Pages i-iii

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2. Introduction

   1. Introduction

      • Johann Pfanzagl

      Pages 1-1

3. Survey of basic theory

   1. Tangent spaces and gradients

      • Johann Pfanzagl

      Pages 2-3

   2. Asymptotic bounds for the concentration of estimator-sequences

      • Johann Pfanzagl

      Pages 4-6

   3. Constructing estimator-sequences

      • Johann Pfanzagl

      Pages 7-16

   4. Estimation in semiparametric models

      • Johann Pfanzagl

      Pages 17-22

   5. Families of gradients

      • Johann Pfanzagl

      Pages 23-34

   6. Estimating equations

      • Johann Pfanzagl

      Pages 35-37

4. Semiparametric families admitting a sufficient statistic

   1. A special semiparametric model

      • Johann Pfanzagl

      Pages 38-47

   2. Mixture models

      • Johann Pfanzagl

      Pages 48-52

   3. Examples of mixture models

      • Johann Pfanzagl

      Pages 53-87

5. Auxiliary results

   1. Auxiliary results

      • Johann Pfanzagl

      Pages 88-105

6. Back Matter

   Pages 106-112

   PDF

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.

• DEX
• boundary element method
• development
• distribution
• eXist
• estimator
• function
• functional
• measure
• median
• probability
• probability measure
• symmetry
• time

• Mathematisches Institut, Universität zu Köln, Köln 41, Federal Republic of Germany

  Johann Pfanzagl

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