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Nonparametric estimators | |
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Examples of nonparametric models and problems | |
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Kernel density estimators | |
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Mean squared error of kernel estimators | |
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Construction of a kernel of order l | |
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Integrated squared risk of kernel estimators | |
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Lack of asymptotic optimality for fixed density | |
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Fourier analysis of kernel density estimators | |
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Unbiased risk estimation. Cross-validation density estimators | |
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Nonparametric regression. The Nadaraya-Watson estimator | |
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Local polynomial estimators | |
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Pointwise and integrated risk of local polynomial estimators | |
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Convergence in the sup-norm | |
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Projection estimators | |
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Sobolev classes and ellipsoids | |
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Integrated squared risk of projection estimators | |
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Generalizations | |
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Oracles | |
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Unbiased risk estimation for regression | |
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Three Gaussian models | |
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Notes | |
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Exercises | |
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Lower bounds on the minimax risk | |
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Introduction | |
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A general reduction scheme | |
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Lower bounds based on two hypotheses | |
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Distances between probability measures | |
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Inequalities for distances | |
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Bounds based on distances | |
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Lower bounds on the risk of regression estimators at a point | |
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Lower bounds based on many hypotheses | |
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Lower bounds in L[subscript 2] | |
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Lower bounds in the sup-norm | |
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Other tools for minimax lower bounds | |
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Fano's lemma | |
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Assouad's lemma | |
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The van Trees inequality | |
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The method of two fuzzy hypotheses | |
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Lower bounds for estimators of a quadratic functional | |
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Notes | |
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Exercises | |
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Asymptotic efficiency and adaptation | |
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Pinsker's theorem | |
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Linear minimax lemma | |
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Proof of Pinsker's theorem | |
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Upper bound on the risk | |
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Lower bound on the minimax risk | |
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Stein's phenomenon | |
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Stein's shrinkage and the James-Stein estimator | |
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Other shrinkage estimators | |
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Superefficiency | |
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Unbiased estimation of the risk | |
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Oracle inequalities | |
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Minimax adaptivity | |
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Inadmissibility of the Pinsker estimator | |
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Notes | |
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Exercises | |
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Appendix | |
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Bibliography | |
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Index | |