Synopsis
I. Information Bounds.- 1 Models, scores, and tangent spaces.- 1.1 Introduction.- 1.2 Models P.- 1.3 Scores: Differentiability of the Model.- 1.4 Tangent Sets P0 and Tangent Spaces P.- 1.5 Score Operators.- 1.6 Exercises.- 2 Convolution and asymptotic minimax theorems.- 2.1 Introduction.- 2.2 Finite-dimensional Parameter Spaces.- 2.3 Infinite-dimensional Parameter Spaces.- 2.4 Exercises.- 3 Van der Vaart's Differentiability Theorem.- 3.1 Differentiability of Implicitly Defined Functions.- 3.2 Some Applications of the Differentiability Theorem.- 3.3 Exercises.- II. Nonparametric Maximum Likelihood Estimation.- 1 The interval censoring problem.- 1.1 Characterization of the non-parametric maximum likelihood estimators.- 1.2Exercises.- 2 The deconvolution problem.- 2.1 Decreasing densities and non-negative random variables.- 2.2 Convolution with symmetric densities.- 2.3 Exercises.- 3 Algorithms.- 3.1 The EM algorithm.- 3.2 The iterative convex minorant algorithm.- 3.3 Exercises.- 4 Consistency.- 4.1 Interval censoring, Case 1.- 4.2 Convolution with a symmetric density.- 4.3 Interval censoring, Case 2.- 4.4 Exercises.- 5 Distribution theory.- 5.1 Interval censoring, Case 1.- 5.2 Interval censoring, Case 2.- 5.3 Deconvolution with a decreasing density.- 5.4 Estimation of the mean.- 5.5 Exercises.- References.
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