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## Book Description

In nonparametric and high-dimensional statistical models, the classical Gauss-Fisher-Le Cam theory of the optimality of maximum likelihood estimators and Bayesian posterior inference does not apply, and new foundations and ideas have been developed in the past several decades. This book gives a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained 'mini-courses' on the theory of Gaussian and empirical processes, on approximation and wavelet theory, and on the basic theory of function spaces. The theory of statistical inference in such models - hypothesis testing, estimation and confidence sets - is then presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, but also Bayesian nonparametrics and nonparametric maximum likelihood estimation. In the final chapter, the theory of adaptive inference in nonparametric models is developed, including Lepski's method, wavelet thresholding, and adaptive inference for self-similar functions.

1. Cover
2. Half-title page
3. Series page
4. Title
6. Dedication
7. Contents
8. Preface
9. 1 Nonparametric Statistical Models
1. 1.1 Statistical Sampling Models
2. 1.2 Gaussian Models
3. 1.3 Notes
10. 2 Gaussian Processes
1. 2.1 Definitions, Separability, 0-1 Law, Concentration
2. 2.2 Isoperimetric Inequalities with Applications to Concentration
3. 2.3 The Metric Entropy Bound for Suprema of Sub-Gaussian Processes
4. 2.4 Anderson’s Lemma, Comparison and Sudakov’s Lower Bound
5. 2.5 The Log-Sobolev Inequality and Further Concentration
6. 2.6 Reproducing Kernel Hilbert Spaces
7. 2.7 Asymptotics for Extremes of Stationary Gaussian Processes
8. 2.8 Notes
11. 3 Empirical Processes
1. 3.1 Definitions, Overview and Some Background Inequalities
3. 3.3 The Entropy Method and Talagrand’s Inequality
4. 3.4 First Applications of Talagrand’s Inequality
5. 3.5 Metric Entropy Bounds for Suprema of Empirical Processes
6. 3.6 Vapnik-Červonenkis Classes of Sets and Functions
7. 3.7 Limit Theorems for Empirical Processes
8. 3.8 Notes
12. 4 Function Spaces and Approximation Theory
1. 4.1 Definitions and Basic Approximation Theory
2. 4.2 Ortho-Normal Wavelet Bases
3. 4.3 Besov Spaces
4. 4.4 Gaussian and Empirical Processes in Besov Spaces
5. 4.5 Notes
13. 5 Linear Nonparametric Estimators
1. 5.1 Kernel and Projection-Type Estimators
2. 5.2 Weak and Multiscale Metrics
3. 5.3 Some Further Topics
4. 5.4 Notes
1. 6.1 Likelihoods and Information
2. 6.2 Testing Nonparametric Hypotheses
3. 6.3 Nonparametric Estimation
4. 6.4 Nonparametric Confidence Sets
5. 6.5 Notes
15. 7 Likelihood-Based Procedures
1. 7.1 Nonparametric Testing in Hellinger Distance
2. 7.2 Nonparametric Maximum Likelihood Estimators
3. 7.3 Nonparametric Bayes Procedures
4. 7.4 Notes