Combinatorial methods in density estimation

1. Introduction.- §1.1. References.- 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.- 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.- 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.- 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.- 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. L2 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.- 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.- 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.- 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.- 10. Additive Estimates and Data Splitting.- §10.1. Data Splitting.- §10.2. Additive Estimates.- §10.3. Histogram Estimates.- §10A. Bibliographic Remarks.- §10.5. Exercises.- §10.6. References.- 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.- 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.- 13. Wavelet Estimates.- §13.1. Definitions.- §13.2. Smoothing.- §13.3. Thresholding.- §13.4. Soft Thresholding.- §13.5. BibliographicRemarks.- §13.6. Exercises.- §13.7. References.- 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.- 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.- 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.- 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.