Wasserman All of Nonparametric Statistics

1. Auflage 2006
ISBN: 978-0-387-30623-0
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 270 Seiten

Reihe: Springer Texts in Statistics

This text provides the reader with a single book where they can find accounts of a number of up-to-date issues in nonparametric inference. The book is aimed at Masters or PhD level students in statistics, computer science, and engineering. It is also suitable for researchers who want to get up to speed quickly on modern nonparametric methods. It covers a wide range of topics including the bootstrap, the nonparametric delta method, nonparametric regression, density estimation, orthogonal function methods, minimax estimation, nonparametric confidence sets, and wavelets. The book's dual approach includes a mixture of methodology and theory.

Autoren/Hrsg.

Wasserman, Larry

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;Introduction;13
3.1;1.1 What Is Nonparametric Inference?;13
3.2;1.2 Notation and Background;14
3.3;1.3 Confidence Sets;17
3.4;1.4 Useful Inequalities;20
3.5;1.5 Bibliographic Remarks;22
3.6;1.6 Exercises;22
4;Estimating the cdf and Statistical Functionals;24
4.1;2.1 The cdf;24
4.2;2.2 Estimating Statistical Functionals;26
4.3;2.3 Influence Functions;29
4.4;2.4 Empirical Probability Distributions;32
4.5;2.5 Bibliographic Remarks;34
4.6;2.6 Appendix;34
4.7;2.7 Exercises;35
5;The Bootstrap and the Jackknife;37
5.1;3.1 The Jackknife;37
5.2;3.2 The Bootstrap;40
5.3;3.3 Parametric Bootstrap;41
5.4;3.4 Bootstrap Confidence Intervals;42
5.5;3.5 Some Theory;45
5.6;3.6 Bibliographic Remarks;47
5.7;3.7 Appendix;47
5.8;3.8 Exercises;49
6;Smoothing: General Concepts;52
6.1;4.1 The Bias–Variance Tradeoff;59
6.2;4.2 Kernels;64
6.3;4.3 Which Loss Function?;66
6.4;4.4 Confidence Sets;66
6.5;4.5 The Curse of Dimensionality;67
6.6;4.6 Bibliographic Remarks;68
6.7;4.7 Exercises;68
7;Nonparametric Regression;70
7.1;5.1 Review of Linear and Logistic Regression;72
7.2;5.2 Linear Smoothers;75
7.3;5.3 Choosing the Smoothing Parameter;77
7.4;5.4 Local Regression;80
7.5;5.5 Penalized Regression, Regularization and Splines;90
7.6;5.6 Variance Estimation;94
7.7;5.7 Confidence Bands;98
7.8;5.8 Average Coverage;103
7.9;5.9 Summary of Linear Smoothing;104
7.10;5.10 Local Likelihood and Exponential Families;105
7.11;5.11 Scale-Space Smoothing;108
7.12;5.12 Multiple Regression;109
7.13;5.13 Other Issues;120
7.14;5.14 Bibliographic Remarks;128
7.15;5.15 Appendix;128
7.16;5.16 Exercises;129
8;Density Estimation;133
8.1;6.1 Cross-Validation;134
8.2;6.2 Histograms;135
8.3;6.3 Kernel Density Estimation;139
8.4;6.4 Local Polynomials;145
8.5;6.5 Multivariate Problems;146
8.6;6.6 Converting Density Estimation Into Regression;147
8.7;6.7 Bibliographic Remarks;148
8.8;6.8 Appendix;148
8.9;6.9 Exercises;150
9;Normal Means and Minimax Theory;153
9.1;7.1 The Normal Means Model;153
9.2;7.2 Function Spaces;155
9.3;7.3 Connection to Regression and Density Estimation;157
9.4;7.4 Stein’s Unbiased Risk Estimator (sure);158
9.5;7.5 Minimax Risk and Pinsker’s Theorem;161
9.6;7.6 Linear Shrinkage and the James–Stein Estimator;163
9.7;7.7 Adaptive Estimation Over Sobolev Spaces;166
9.8;7.8 Confidence Sets;167
9.9;7.9 Optimality of Confidence Sets;174
9.10;7.10 Random Radius Bands?;178
9.11;7.11 Penalization, Oracles and Sparsity;179
9.12;7.12 Bibliographic Remarks;180
9.13;7.13 Appendix;181
9.14;7.14 Exercises;188
10;Nonparametric Inference Using Orthogonal Functions;191
10.1;8.1 Introduction;191
10.2;8.2 Nonparametric Regression;191
10.3;8.3 Irregular Designs;198
10.4;8.4 Density Estimation;200
10.5;8.5 Comparison of Methods;201
10.6;8.6 Tensor Product Models;201
10.7;8.7 Bibliographic Remarks;202
10.8;8.8 Exercises;202
11;Wavelets and Other Adaptive Methods;204
11.1;9.1 Haar Wavelets;206
11.2;9.2 Constructing Wavelets;210
11.3;9.3 Wavelet Regression;213
11.4;9.4 Wavelet Thresholding;215
11.5;9.5 Besov Spaces;218
11.6;9.6 Confidence Sets;221
11.7;9.7 Boundary Corrections and Unequally Spaced Data;222
11.8;9.8 Overcomplete Dictionaries;222
11.9;9.9 Other Adaptive Methods;223
11.10;9.10 Do Adaptive Methods Work?;227
11.11;9.11 Bibliographic Remarks;228
11.12;9.12 Appendix;228
11.13;9.13 Exercises;230
12;Other Topics;233
12.1;10.1 Measurement Error;233
12.2;10.2 Inverse Problems;239
12.3;10.3 Nonparametric Bayes;241
12.4;10.4 Semiparametric Inference;241
12.5;10.5 Correlated Errors;242
12.6;10.6 Classification;242
12.7;10.7 Sieves;243
12.8;10.8 Shape-Restricted Inference;243
12.9;10.9 Testing;244
12.10;10.10 Computational Issues;246
12.11;10.11 Exercises;246
13;Bibliography;249
14;Index;266