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Table of contents
1 Introduction
1.1 Why set estimation ?
1.2 Definitions and Notation
1.2.1 Set-valued estimators
1.2.2 Notation and first definitions
1.2.3 The minimax setup
1.2.4 Adaptation and misspecification
1.3 The statistical models
2 Estimation of the convex or polytopal support of a uniform density
2.1 Random polytopes
2.1.1 The wet part and the floating convex body
2.1.2 Convergence of the random convex hull
2.1.3 Asymptotic facial structure
2.2 Minimax estimation
2.2.1 Estimation of polytopes
2.2.2 Estimation of convex bodies
2.2.3 Adaptation and misspecification
2.3 Further results and open problems
2.3.1 A universal deviation inequality for the random convex hull
2.3.2 Moments of the number vertices of the convex hull estimator
2.3.3 Efron’s identity revisited
2.3.4 Non uniform distributions
2.3.5 Estimation of log-concave densities
2.4 Proofs
2.5 Appendix to Chapter 2
2.5.1 Lemmata
2.5.2 Proofs of Lemmata and others
3 A nonparametric regression model
3.1 Estimation of polytopes
3.1.1 Upper bound
3.1.2 Lower bound
3.2 Estimation of convex bodies
3.3 Adaptation and misspecification
3.4 Discussion
3.5 The one-dimensional case and the change point problem
3.5.1 Detection of a segment
3.5.2 Estimation of a segment
3.5.3 Conclusion and discussion
3.6 Proofs
3.7 Appendix to Chapter 3
3.7.1 Lemmata
3.7.2 Proof of the lemmata
4 Estimation of functionals
4.1 The density support model
4.1.1 Estimation of the volume of a convex body
4.1.2 Estimation of the area of a triangle
4.1.3 Estimation of intrinsic volumes
4.2 The regression model
5 Conclusion, contributions
