Combinatorial Methods in Density Estimation
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1 Luc Devroye Gabor Lugosi Combinatorial Methods in Density Estimation Springer
2 Contents Preface vii 1. Introduction 1 a 1.1. References 3 2. Concentration Inequalities Hoeffding's Inequality An Inequality for the Expected Maximal Deviation The Bounded Difference Inequality 7 y 2.4. Examples Bibliographic Remarks Exercises References Uniform Deviation Inequalities The Vapnik-Chervonenkis Inequality Covering Numbers and Chaining Example: The Dvoretzky-Kiefer-Wolfowitz Theorem Bibliographic Remarks Exercises References Combinatorial Tools Shatter Coefficients Vapnik-Chervonenkis Dimension and Shatter Coefficients Vapnik-Chervonenkis Dimension and Covering Numbers Examples Bibliographic Remarks Exercises References Total Variation Density Estimation The Total Variation^ Invariance Mappings Convolutions Normalization The Lebesgue Density Theorem 42
3 x/ CONTENTS 5.8. LeCam's Inequality Bibliographic Remarks Exercises References Choosing a Density Estimate Choosing Between Two Densities Examples Is the Factor of Three Necessary? Maximum Likelihood Does not Work *2 Distances Are To Be Avoided Selection from A; Densities Examples Continued Selection from an Infinite Class Bibliographic Remarks Exercises References Skeleton Estimates Kolmogorov Entropy Skeleton Estimates Robustness Finite Mixtures Monotone Densities on the Hypercube How To Make Gigantic Totally Bounded Classes Bibliographic Remarks 66 ' 7.8. Exercises References The Minimum Distance Estimate: Examples Problem Formulation Series Estimates Parametric Estimates: Exponential Families Neural Network Estimates Mixture Classes, Radial Basis Function Networks Bibliographic Remarks Exercises References The Kernel Density Estimate Approximating Functions by Convolutions Definition of the Kernel Estimate Consistency of the Kernel Estimate Concentration Choosing the Bandwidth Choosing the Kernel Rates of Convergence 85
4 CONTENTS / Xi 9.8. Uniform Rate of Convergence Shrinkage, and the Combination of Density Estimates Bibliographic Remarks Exercises References Additive Estimates and Data Splitting Data Splitting Additive Estimates Histogram Estimates Bibliographic Remarks Exercises References Bandwidth Selection for Kernel Estimates The Kernel Estimate with Riemann Kernel General Kernels, Kernel Complexity Kernel Complexity: Univariate Examples Kernel Complexity: Multivariate Kernels Asymptotic Optimality Bibliographic Remarks Exercises References Multiparameter Kernel Estimates Multivariate Kernel Estimates Product Kernels 118, Multivariate Kernel Estimates Ellipsoidal Kernels Variable Kernel Estimates Tree-Structured Partitions Changepoints and Bump Hunting Bibliographic Remarks Exercises References Wavelet Estimates Definitions Smoothing Thresholding Soft Thresholding Bibliographic Remarks Exercises References The Transformed Kernel Estimate The Transformed Kernel Estimate Box-Cox Transformations Piecewise Linear Transformations Bibliographic Remarks 148
5 xii/ CONTENTS Exercises References Minimax Theory Estimating a Density from One Data Point The General Minimax Problem Rich Classes Assouad's Lemma ^Example: The Class of Convex Densities Additional Examples Tuning the Parameters of Variable Kernel Estimates Sufficient Statistics Bibliographic Remarks Exercises References Choosing the Kernel Order Introduction Standard Kernel Estimate: Riemann Kernels Standard Kernel Estimates: General Kernels An Infinite Family of Kernels Bibliographic Remarks Exercises References Bandwidth Choice with SuperMerlieis Superkernels The Trapezoidal Kernel Bandwidth Selection Bibliographic Remarks Exercises References 196 Author Index 199 Subject Index 203
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