New book by Luc Devroye and Gabor Lugosi Springer-Verlag, New York, 2001 ISBN number 0-387-95117-2 Available at Amazon.com for 44.95USD Combinatorial Methods in Density Estimation

Preface

Chapter 1

Introduction

Chapter 2

Concentration Inequalities 2.1. Hoeffding's Inequality 2.2. An Inequality for the Expected Maximal Deviation 2.3. The Bounded Difference Inequality 2.4. Examples 2.5. Bibliographic Remarks 2.6. Exercises 2.7. References

Chapter 3

Uniform Deviation Inequalities 3.1. The Vapnik--Chervonenkis Inequality 3.2. Covering Numbers and Chaining 3.3. Example: The Dvoretzky--Kiefer--Wolfowitz Theorem 3.4. Bibliographic Remarks 3.5. Exercises 3.6. References

Chapter 4

Combinatorial Tools 4.1. Shatter Coefficients 4.2. Vapnik--Chervonenkis Dimension and Shatter Coefficients 4.3. Vapnik--Chervonenkis Dimension and Covering Numbers 4.4. Examples 4.5. Bibliographic remarks 4.6. Exercises 4.7. References

Chapter 5

Total Variation 5.1. Density Estimation 5.2. The Total Variation 5.3. Invariance 5.4. Mappings 5.5. Convolutions 5.6. Normalization 5.7. The Lebesgue Density Theorem 5.8. LeCam's Inequality 5.9. Bibliographic Remarks 5.10. Exercises 5.11. References

Chapter 6

Choosing a Density Estimate 6.1. Choosing Between Two Densities 6.2. Examples 6.3. Is the Factor of Three Necessary? 6.4. Maximum Likelihood Does not Work 6.5. $L_2$ Distances Are To Be Avoided 6.6. Selection from $k$ Densities 6.7. Examples Continued 6.8. Selection from an Infinite Class 6.9. Bibliographic Remarks 6.10. Exercises 6.11. References

Chapter 7

Skeleton Estimates 7.1. Kolmogorov Entropy 7.2. Skeleton Estimates 7.3. Robustness 7.4. Finite Mixtures 7.5. Monotone Densities on the Hypercube 7.6. How To Make Gigantic Totally Bounded Classes 7.7. Bibliographic Remarks 7.8. Exercises 7.9. References

Chapter 8

The Minimum Distance Estimate: Examples 8.1. Problem Formulation 8.2. Series Estimates 8.3. Parametric Estimates: Exponential Families 8.4. Neural Network Estimates 8.5. Mixture Classes, Radial Basis Function Networks 8.6. Bibliographic Remarks 8.7. Exercises 8.8. References

Chapter 9

The Kernel Density Estimate 9.1. Approximating Functions by Convolutions 9.2. Definition of the Kernel Estimate 9.3. Consistency of the Kernel Estimate 9.4. Concentration 9.5. Choosing the Bandwidth 9.6. Choosing the Kernel 9.7. Rates of Convergence 9.8. Uniform Rate of Convergence 9.9. Shrinkage, and the Combination of Density Estimates 9.10. Bibliographic Remarks 9.11. Exercises 9.12. References

Chapter 10

Additive Estimates and Data Splitting 10.1. Data Splitting 10.2. Additive Estimates 10.3. Histogram Estimates 10.4. Bibliographic Remarks 10.5. Exercises 10.6. References

Chapter 11

Bandwidth Selection for Kernel Estimates 11.1. The Kernel Estimate with Riemann Kernel 11.2. General Kernels, Kernel Complexity 11.3. Kernel Complexity: Univariate Examples 11.4. Kernel Complexity: Multivariate Kernels 11.5. Asymptotic Optimality 11.6. Bibliographic Remarks 11.7. Exercises 11.8. References

Chapter 12

Multiparameter Kernel Estimates 12.1. Multivariate Kernel Estimates---Product Kernels 12.2. Multivariate Kernel Estimates---Ellipsoidal Kernels 12.3. Variable Kernel Estimates 12.4. Tree-Structured Partitions 12.5. Changepoints and Bump Hunting 12.6. Bibliographic Remarks 12.7. Exercises 12.8. References

Chapter 13

Wavelet Estimates 13.1. Definitions 13.2. Smoothing 13.3. Thresholding 13.4. Soft Thresholding 13.5. Bibliographic Remarks 13.6. Exercises 13.7. References

Chapter 14

The Transformed Kernel Estimate 14.1. The Transformed Kernel Estimate 14.2. Box--Cox Transformations 14.3. Piecewise Linear Transformations 14.4. Bibliographic Remarks 14.5. Exercises 14.6. References

Chapter 15

Minimax Theory 15.1. Estimating a Density from One Data Point 15.2. The General Minimax Problem 15.3. Rich Classes 15.4. Assouad's Lemma 15.5. Example: The Class of Convex Densities 15.6. Additional Examples 15.7. Tuning the Parameters of Variable Kernel Estimates 15.8. Sufficient Statistics 15.9. Bibliographic Remarks 15.10. Exercises 15.11. References

Chapter 16

Choosing the Kernel Order 16.1. Introduction 16.2. Standard Kernel Estimate: Riemann Kernels 16.3. Standard Kernel Estimates: General Kernels 16.4. An Infinite Family of Kernels 16.5. Bibliographic Remarks 16.6. Exercises 16.7. References

Chapter 17

Bandwidth Choice with Superkernels 17.1. Superkernels 17.2. The Trapezoidal Kernel 17.3. Bandwidth Selection 17.4. Bibliographic Remarks 17.5. Exercises 17.6. References

Author Index

Subject Index

Copyright © 2000 Luc Devroye. Email: luc@cs.mcgill.ca.