Information Bounds and Nonparametric Maximum Likelihood Estimation

1. Front Matter

   Pages i-viii

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2. Information Bounds

   1. Front Matter

      Pages 1-1

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   2. Models, Scores, and Tangent Spaces

      • Piet Groeneboom, Jon A. Wellner

      Pages 3-12

   3. Convolution and Asymptotic Minimax Theorems

      • Piet Groeneboom, Jon A. Wellner

      Pages 13-21

   4. Van der Vaart’s Differentiability Theorem

      • Piet Groeneboom, Jon A. Wellner

      Pages 23-32

3. Nonparametric Maximum Likelihood Estimation

   1. Front Matter

      Pages 33-33

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   2. The Interval Censoring Problem

      • Piet Groeneboom, Jon A. Wellner

      Pages 35-52

   3. The Deconvolution Problem

      • Piet Groeneboom, Jon A. Wellner

      Pages 53-63

   4. Algorithms

      • Piet Groeneboom, Jon A. Wellner

      Pages 65-74

   5. Consistency

      • Piet Groeneboom, Jon A. Wellner

      Pages 75-87

   6. Distribution Theory

      • Piet Groeneboom, Jon A. Wellner

      Pages 89-121

4. Back Matter

   Pages 123-126

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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.

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

• Dept. of Mathematics and Computer Science, Delft University of Technology, Delft, Netherlands

  Piet Groeneboom

• Dept. of Statistics GN-22, University of Washington, Seattle, USA

  Jon A. Wellner

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