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AuthorDevroye, Luc. author
TitleCombinatorial Methods in Density Estimation [electronic resource] / by Luc Devroye, Gรกbor Lugosi
ImprintNew York, NY : Springer New York : Imprint: Springer, 2001
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Descript XII, 209 p. online resource


Density estimation has evolved enormously since the days of bar plots and histograms, but researchers and users are still struggling with the problem of the selection of the bin widths. This text explores a new paradigm for the data-based or automatic selection of the free parameters of density estimates in general so that the expected error is within a given constant multiple of the best possible error. The paradigm can be used in nearly all density estimates and for most model selection problems, both parametric and nonparametric. It is the first book on this topic. The text is intended for first-year graduate students in statistics and learning theory, and offers a host of opportunities for further research and thesis topics. Each chapter corresponds roughly to one lecture, and is supplemented with many classroom exercises. A one year course in probability theory at the level of Feller's Volume 1 should be more than adequate preparation. Gabor Lugosi is Professor at Universitat Pompeu Fabra in Barcelona, and Luc Debroye is Professor at McGill University in Montreal. In 1996, the authors, together with Lรกszlo Gyรถrfi, published the successful text, A Probabilistic Theory of Pattern Recognition with Springer-Verlag. Both authors have made many contributions in the area of nonparametric estimation


1. Introduction -- ยง1.1. References -- 2. Concentration Inequalities -- ยง2.1. Hoeffdingโ{128}{153}s Inequality -- ยง2.2. An Inequality for the Expected Maximal Deviation -- ยง2.3. The Bounded Difference Inequality -- ยง2.4. Examples -- ยง2.5. Bibliographic Remarks -- ยง2.6. Exercises -- ยง2.7. References -- 3. Uniform Deviation Inequalities -- ยง3.1. The Vapnik-Chervonenkis Inequality -- ยง3.2. Covering Numbers and Chaining -- ยง3.3. Example: The Dvoretzky-Kiefer-Wolfowitz Theorem -- ยง3.4. Bibliographic Remarks -- ยง3.5. Exercises -- ยง3.6. References -- 4. Combinatorial Tools -- ยง4.1. Shatter Coefficients -- ยง4.2. Vapnik-Chervonenkis Dimension and Shatter Coefficients -- ยง4.3. Vapnik-Chervonenkis Dimension and Covering Numbers -- ยง4.4. Examples -- ยง4.5. Bibliographic Remarks -- ยง4.6. Exercises -- ยง4.7. References -- 5. Total Variation -- ยง5.1. Density Estimation -- ยง5.2. The Total Variation -- ยง5.3. Invariance -- ยง5.4. Mappings -- ยง5.5. Convolutions -- ยง5.6. Normalization -- ยง5.7. The Lebesgue Density Theorem -- ยง5.8. LeCamโ{128}{153}s Inequality -- ยง5.9. Bibliographic Remarks -- ยง5.10. Exercises -- ยง5.11. References -- 6. Choosing a Density Estimate -- ยง6.1. Choosing Between Two Densities -- ยง6.2. Examples -- ยง6.3. Is the Factor of Three Necessary? -- ยง6.4. Maximum Likelihood Does not Work -- ยง6.5. L2 Distances Are To Be Avoided -- ยง6.6. Selection from k Densities -- ยง6.7. Examples Continued -- ยง6.8. Selection from an Infinite Class -- ยง6.9. Bibliographic Remarks -- ยง6.10. Exercises -- ยง6.11. References -- 7. Skeleton Estimates -- ยง7.1. Kolmogorov Entropy -- ยง7.2. Skeleton Estimates -- ยง7.3. Robustness -- ยง7.4. Finite Mixtures -- ยง7.5. Monotone Densities on the Hypercube -- ยง7.6. How To Make Gigantic Totally Bounded Classes -- ยง7.7. Bibliographic Remarks -- ยง7.8. Exercises -- ยง7.9. References -- 8. The Minimum Distance Estimate: Examples -- ยง8.1. Problem Formulation -- ยง8.2. Series Estimates -- ยง8.3. Parametric Estimates: Exponential Families -- ยง8.4. Neural Network Estimates -- ยง8.5. Mixture Classes, Radial Basis Function Networks -- ยง8.6. Bibliographic Remarks -- ยง8.7. Exercises -- ยง8.8. References -- 9. The Kernel Density Estimate -- ยง9.1. Approximating Functions by Convolutions -- ยง9.2. Definition of the Kernel Estimate -- ยง9.3. Consistency of the Kernel Estimate -- ยง9.4. Concentration -- ยง9.5. Choosing the Bandwidth -- ยง9.6. Choosing the Kernel -- ยง9.7. Rates of Convergence -- ยง9.8. Uniform Rate of Convergence -- ยง9.9. Shrinkage, and the Combination of Density Estimates -- ยง9.10. Bibliographic Remarks -- ยง9.11. Exercises -- ยง9.12. References -- 10. Additive Estimates and Data Splitting -- ยง10.1. Data Splitting -- ยง10.2. Additive Estimates -- ยง10.3. Histogram Estimates -- ยง10A. Bibliographic Remarks -- ยง10.5. Exercises -- ยง10.6. References -- 11. Bandwidth Selection for Kernel Estimates -- ยง11.1. The Kernel Estimate with Riemann Kernel -- ยง11.2. General Kernels, Kernel Complexity -- ยง11.3. Kernel Complexity: Univariate Examples -- ยง11.4. Kernel Complexity: Multivariate Kernels -- ยง11.5. Asymptotic Optimality -- ยง11.6. Bibliographic Remarks -- ยง11.7. Exercises -- ยง11.8. References -- 12. Multiparameter Kernel Estimates -- ยง12.1. Multivariate Kernel Estimatesโ{128}{148}Product Kernels -- ยง12.2. Multivariate Kernel Estimatesโ{128}{148}Ellipsoidal Kernels -- ยง12.3. Variable Kernel Estimates -- ยง12.4. Tree-Structured Partitions -- ยง12.5. Changepoints and Bump Hunting -- ยง12.6. Bibliographic Remarks -- ยง12.7. Exercises -- ยง12.8. References -- 13. Wavelet Estimates -- ยง13.1. Definitions -- ยง13.2. Smoothing -- ยง13.3. Thresholding -- ยง13.4. Soft Thresholding -- ยง13.5. Bibliographic Remarks -- ยง13.6. Exercises -- ยง13.7. References -- 14. The Transformed Kernel Estimate -- ยง14.1. The Transformed Kernel Estimate -- ยง14.2. Box-Cox Transformations -- ยง14.3. Piecewise Linear Transformations -- ยง14.4. Bibliographic Remarks -- ยง14.5. Exercises -- ยง14.6. References -- 15. Minimax Theory -- ยง15.1. Estimating a Density from One Data Point -- ยง15.2. The General Minimax Problem -- ยง15.3. Rich Classes -- ยง15.4. Assouadโ{128}{153}s Lemma -- ยง15.5. Example: The Class of Convex Densities -- ยง15.6. Additional Examples -- ยง15.7. Tuning the Parameters of Variable Kernel Estimates -- ยง15.8. Sufficient Statistics -- ยง15.9. Bibliographic Remarks -- ยง15.10. Exercises -- ยง15.11. References -- 16. Choosing the Kernel Order -- ยง16.1. Introduction -- ยง16.2. Standard Kernel Estimate: Riemann Kernels -- ยง16.3. Standard Kernel Estimates: General Kernels -- ยง16.4. An Infinite Family of Kernels -- ยง16.5. Bibliographic Remarks -- ยง16.6. Exercises -- ยง16.7. References -- 17. Bandwidth Choice with Superkernels -- ยง17.1. Superkernels -- ยง17.2. The Trapezoidal Kernel -- ยง17.3. Bandwidth Selection -- ยง17.4. Bibliographic Remarks -- ยง17.5. Exercises -- ยง17.6. References -- Author Index

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