Combinatorial methods in density estimation:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2001
|
| Series: | Springer series in statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XII, 208 S. 25 cm |
| ISBN: | 0387951172 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface vii
1. Introduction 1
§1.1. References 3
2. Concentration Inequalities 4
§2.1. Hoeffding s Inequality 4
§2.2. An Inequality for the Expected Maximal Deviation 7
§2.3. The Bounded Difference Inequality 7
§2.4. Examples 9
§2.5. Bibliographic Remarks 10
§2.6. Exercises 11
§2.7. References 13
3. Uniform Deviation Inequalities 17
§3.1. The Vapnik Chervonenkis Inequality 17
§3.2. Covering Numbers and Chaining 19
§3.3. Example: The Dvoretzky Kiefer Wolfowitz Theorem 22
§3.4. Bibliographic Remarks 23
§3.5. Exercises 23
§3.6. References 25
4. Combinatorial Tools 27
§4.1. Shatter Coefficients 27
§4.2. Vapnik Chervonenkis Dimension and Shatter Coefficients 28
§4.3. Vapnik Chervonenkis Dimension and Covering Numbers 30
§4.4. Examples 31
§4.5. Bibliographic Remarks 33
§4.6. Exercises 33
§4.7. References 35
5. Total Variation 38
§5.1. Density Estimation 38
§5.2. The Total Variation 39
§5.3. Invariance 39
§5.4. Mappings 40
§5.5. Convolutions 41
§5.6. Normalization 41
§5.7. The Lebesgue Density Theorem 42
x/ CONTENTS
§5.8. LeCam s Inequality 43
§5.9. Bibliographic Remarks 43
§5.10. Exercises 43
§5.11. References 46
6. Choosing a Density Estimate 47
§6.1. Choosing Between Two Densities 47
§6.2. Examples 49
§6.3. Is the Factor of Three Necessary? 51
§6.4. Maximum Likelihood Does not Work 52
§6.5. Li Distances Are To Be Avoided 52
§6.6. Selection from k Densities 53
§6.7. Examples Continued 55
§6.8. Selection from an Infinite Class 55
§6.9. Bibliographic Remarks 56
§6.10. Exercises 56
§6.11. References 57
7. Skeleton Estimates 58
§7.1. Kolmogorov Entropy 58
§7.2. Skeleton Estimates 58
§7.3. Robustness 60
§7.4. Finite Mixtures 60
§7.5. Monotone Densities on the Hypercube 61
§7.6. How To Make Gigantic Totally Bounded Classes 64
§7.7. Bibliographic Remarks 66
§7.8. Exercises 66
§7.9. References 68
8. The Minimum Distance Estimate: Examples 70
§8.1. Problem Formulation 70
§8.2. Series Estimates 71
§8.3. Parametric Estimates: Exponential Families 72
§8.4. Neural Network Estimates 73
§8.5. Mixture Classes, Radial Basis Function Networks 74
§8.6. Bibliographic Remarks 76
§8.7. Exercises 76
§8.8. References 77
9. The Kernel Density Estimate 79
§9.1. Approximating Functions by Convolutions 79
§9.2. Definition of the Kernel Estimate 80
§9.3. Consistency of the Kernel Estimate 81
§9.4. Concentration 82
§9.5. Choosing the Bandwidth 83
§9.6. Choosing the Kernel 84
§9.7. Rates of Convergence 85
CONTENTS / xi
§9.8. Uniform Rate of Convergence 86
§9.9. Shrinkage, and the Combination of Density Estimates 88
§9.10. Bibliographic Remarks 90
§9.11. Exercises 90
§9.12. References 95
10. Additive Estimates and Data Splitting 98
§10.1. Data Splitting 98
§10.2. Additive Estimates 99
§10.3. Histogram Estimates 103
§10.4. Bibliographic Remarks 105
§10.5. Exercises 105
§10.6. References 107
11. Bandwidth Selection for Kernel Estimates 108
§11.1. The Kernel Estimate with Riemann Kernel 108
§11.2. General Kernels, Kernel Complexity 110
§11.3. Kernel Complexity: Univariate Examples 111
§11.4. Kernel Complexity: Multivariate Kernels 113
§11.5. Asymptotic Optimality 114
§11.6. Bibliographic Remarks 115
§11.7. Exercises 115
§11.8. References 116
12. Multiparameter Kernel Estimates 118
§12.1. Multivariate Kernel Estimates—Product Kernels 118
§12.2. Multivariate Kernel Estimates—Ellipsoidal Kernels 121
§12.3. Variable Kernel Estimates 122
§12.4. Tree Structured Partitions 124
§12.5. Changepoints and Bump Hunting 125
§12.6. Bibliographic Remarks 127
§12.7. Exercises 127
§12.8. References 132
13. Wavelet Estimates 134
§13.1. Definitions 134
§13.2. Smoothing 135
§13.3. Thresholding 136
§13.4. Soft Thresholding 138
§13.5. Bibliographic Remarks 139
§13.6. Exercises 139
§13.7. References 140
14. The Transformed Kernel Estimate 142
§14.1. The Transformed Kernel Estimate 142
§14.2. Box Cox Transformations 143
§14.3. Piecewise Linear Transformations 146
§14.4. Bibliographic Remarks 148
xii/ CONTENTS
§14.5. Exercises 148
§14.6. References 149
15. Minimax Theory 150
§15.1. Estimating a Density from One Data Point 150
§15.2. The General Minimax Problem 152
§15.3. Rich Glasses 154
§15.4. Assouad s Lemma 156
§15.5. Example: The Class of Convex Densities 159
§15.6. Additional Examples 162
§15.7. Tuning the Parameters of Variable Kernel Estimates 163
§15.8. Sufficient Statistics 166
§15.9. Bibliographic Remarks 168
§15.10. Exercises 169
§15.11. References 174
16. Choosing the Kernel Order 177
§16.1. Introduction 177
§16.2. Standard Kernel Estimate: Riemann Kernels 179
§16.3. Standard Kernel Estimates: General Kernels 181
§16.4. An Infinite Family of Kernels 184
§16.5. Bibliographic Remarks 187
§16.6. Exercises 188
§16.7. References 188
17. Bandwidth Choice with Superkernels 190
§17.1. Superkernels 190
§17.2. The Trapezoidal Kernel 192
§17.3. Bandwidth Selection 193
§17.4. Bibliographic Remarks 194
§17.5. Exercises 194
§17.6. References 196
Author Index 199
Subject Index 203
|
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-search | 512.2/4 21 512.2/4 |
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| dewey-tens | 510 - Mathematics |
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| spelling | Devroye, Luc Verfasser (DE-588)170228444 aut Combinatorial methods in density estimation Luc Devroye ; Gábor Lugosi New York [u.a.] Springer 2001 XII, 208 S. 25 cm txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Literaturangaben Analyse combinatoire Combinatieleer gtt Distribution (Théorie des probabilités) Estimation, Théorie de l' Estimação de densidades larpcal Inferência estatística larpcal Schattingstheorie gtt Verdelingen (statistiek) gtt Distribution (Probability theory) Estimation theory Combinatorial analysis Dichteschätzung (DE-588)4353528-8 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 s Dichteschätzung (DE-588)4353528-8 s DE-604 Lugosi, Gábor 1964- Verfasser (DE-588)17173677X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009346038&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Devroye, Luc Lugosi, Gábor 1964- Combinatorial methods in density estimation Analyse combinatoire Combinatieleer gtt Distribution (Théorie des probabilités) Estimation, Théorie de l' Estimação de densidades larpcal Inferência estatística larpcal Schattingstheorie gtt Verdelingen (statistiek) gtt Distribution (Probability theory) Estimation theory Combinatorial analysis Dichteschätzung (DE-588)4353528-8 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd |
| subject_GND | (DE-588)4353528-8 (DE-588)4121894-2 |
| title | Combinatorial methods in density estimation |
| title_auth | Combinatorial methods in density estimation |
| title_exact_search | Combinatorial methods in density estimation |
| title_full | Combinatorial methods in density estimation Luc Devroye ; Gábor Lugosi |
| title_fullStr | Combinatorial methods in density estimation Luc Devroye ; Gábor Lugosi |
| title_full_unstemmed | Combinatorial methods in density estimation Luc Devroye ; Gábor Lugosi |
| title_short | Combinatorial methods in density estimation |
| title_sort | combinatorial methods in density estimation |
| topic | Analyse combinatoire Combinatieleer gtt Distribution (Théorie des probabilités) Estimation, Théorie de l' Estimação de densidades larpcal Inferência estatística larpcal Schattingstheorie gtt Verdelingen (statistiek) gtt Distribution (Probability theory) Estimation theory Combinatorial analysis Dichteschätzung (DE-588)4353528-8 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd |
| topic_facet | Analyse combinatoire Combinatieleer Distribution (Théorie des probabilités) Estimation, Théorie de l' Estimação de densidades Inferência estatística Schattingstheorie Verdelingen (statistiek) Distribution (Probability theory) Estimation theory Combinatorial analysis Dichteschätzung Wahrscheinlichkeitsverteilung |
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