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Information Bounds and Nonparametric Maximum Likelihood Estimation

  • Piet Groeneboom
  • Jon A. Wellner

Part of the DMV Seminar book series (OWS, volume 19)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Information Bounds

    1. Front Matter
      Pages 1-1
    2. Piet Groeneboom, Jon A. Wellner
      Pages 3-12
    3. Piet Groeneboom, Jon A. Wellner
      Pages 13-21
    4. Piet Groeneboom, Jon A. Wellner
      Pages 23-32
  3. Nonparametric Maximum Likelihood Estimation

    1. Front Matter
      Pages 33-33
    2. Piet Groeneboom, Jon A. Wellner
      Pages 35-52
    3. Piet Groeneboom, Jon A. Wellner
      Pages 53-63
    4. Piet Groeneboom, Jon A. Wellner
      Pages 65-74
    5. Piet Groeneboom, Jon A. Wellner
      Pages 75-87
    6. Piet Groeneboom, Jon A. Wellner
      Pages 89-121
  4. Back Matter
    Pages 123-126

About this book

Introduction

This book contains the lecture notes for a DMV course presented by the authors at Gunzburg, Germany, in September, 1990. In the course we sketched the theory of information bounds for non parametric and semiparametric models, and developed the theory of non parametric maximum likelihood estimation in several particular inverse problems: interval censoring and deconvolution models. Part I, based on Jon Wellner's lectures, gives a brief sketch of information lower bound theory: Hajek's convolution theorem and extensions, useful minimax bounds for parametric problems due to Ibragimov and Has'minskii, and a recent result characterizing differentiable functionals due to van der Vaart (1991). The differentiability theorem is illustrated with the examples of interval censoring and deconvolution (which are pursued from the estimation perspective in part II). The differentiability theorem gives a way of clearly distinguishing situations in which 1 2 the parameter of interest can be estimated at rate n / and situations in which this is not the case. However it says nothing about which rates to expect when the functional is not differentiable. Even the casual reader will notice that several models are introduced, but not pursued in any detail; many problems remain. Part II, based on Piet Groeneboom's lectures, focuses on non parametric maximum likelihood estimates (NPMLE's) for certain inverse problems. The first chapter deals with the interval censoring problem.

Keywords

Censoring Estimator Finite Likelihood Random variable Variable expectation–maximization algorithm function theorem

Authors and affiliations

  • Piet Groeneboom
    • 1
  • Jon A. Wellner
    • 2
  1. 1.Dept. of Mathematics and Computer ScienceDelft University of TechnologyDelftNetherlands
  2. 2.Dept. of Statistics GN-22University of WashingtonSeattleUSA

Bibliographic information