دانلود کتاب Weak Convergence and Empirical Processes: With Applications to Statistics (Springer Series in Statistics)

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Preface
Preface to the Second Edition
Reading Guide
Contents
PART 1 Stochastic Convergence
1.1 Introduction
1.2 Outer Integrals and Measurable Majorants
Problems and Complements
1.3 Weak Convergence
Problems and Complements
1.4 Product Spaces
1.5 Spaces of Bounded Functions
Problems and Complements
1.6 Spaces of Locally Bounded Functions
1.6.1 Convex processes
Problems and Complements
1.7 Ball Sigma-Field and Measurability of Suprema
Problems and Complements
1.8 Hilbert Spaces
Problems and Complements
1.9 Almost Sure and in Probability Convergence
Problems and Complements
1.10 Weak, Almost Uniform and in Probability Convergence
Problems and Complements
1.11 Refinements
Problems and Complements
1.12 Uniformity and Metrization
Problems and Complements
1.13 Conditional Weak Convergence
Problems and Complements
1.14 Skorokhod Space
1.14.1 Compact Domain
1.14.2 Unbounded Domain
1.14.2.1 Conditional Distribution Bounds
1.14.2.2 Mixed Moment Bounds
1.14.2.3 Counting Processes
1.14.2.4 Semi-Martingales
Problems and Complements
1.15 Stable Convergence
Problems and Complements
1 Notes
PART 2 Empirical Processes
2.1 Introduction
2.1.1 Overview
2.1.2 Asymptotic Equicontinuity
2.1.3 Maximal Inequalities
2.1.4 Central Limit Theorem in Banach Spaces
Problems and Complements
2.2 Maximal Inequalities
2.2.1 Orlicz Norms
2.2.2 Covering Numbers and Entropy
2.2.3 Sub-Gaussian Inequalities
2.2.4 Mixed Sub-Gaussian-Exponential Inequalities
2.2.5 Generic Chaining
2.2.6 Majorizing Measures
Problems and Complements
2.3 Symmetrization and Measurability
2.3.1 Symmetrization
2.3.2 More Symmetrization
2.3.3 Separable Versions
Problems and Complements
2.4 Glivenko-Cantelli Theorems
Problems and Complements
2.5 Donsker Theorems
2.5.1 Uniform Entropy
2.5.2 Bracketing
Problems and Complements
2.6 Uniform Entropy Numbers
2.6.1 VC-Classes of Sets
2.6.2 VC-Classes of Functions
2.6.3 Convex Hulls and VC-Hull Classes
2.6.4 VC-Major Classes
2.6.5 Examples
2.6.6 Permanence Properties
2.6.7 Fat Shattering
Problems and Complements
2.7 Entropies of Function Classes
2.7.1 Hölder Classes
2.7.2 Sobolev and Besov Spaces
2.7.2.1 Besov Bodies
2.7.3 Monotone Functions
2.7.4 Convex Sets and Functions
2.7.5 Analytic Functions
2.7.6 Parametrized Classes
2.7.7 Gaussian Mixtures
2.7.8 Sets with Smooth Boundaries
Problems and Complements
2.8 Uniformity in the Underlying Distribution
2.8.1 Glivenko-Cantelli Theorems
2.8.2 Donsker Theorems
2.8.3 Central Limit Theorem Under Sequences
Problems and Complements
2.9 Multiplier Central Limit Theorems
Problems and Complements
2.10 Permanence under Transformation
2.10.1 Closures and Convex Hulls
2.10. 2Transformations of Glivenko-Cantelli Classes
2.10.3 Transformations of Donsker Classes
2.10.4 Uniform Entropy under Transformation
2.10.5 Partitioning the Sample Space
Problems and Complements
2.11 Central Limit Theorem for Processes
2.11.1 Random Entropy
2.11.1.1 Measurelike Processes
2.11.2 Bracketing
2.11.3 Changing Classes of Functions
Problems and Complements
2.12 Partial-Sum Processes
2.12.1 Sequential Empirical Process
2.12.2 Partial-Sum Processes on Lattices
Problems and Complements
2.13 Other Donsker Classes
2.13.1 Sequences
2.13.2 Elliptical Classes
2.13.3 Classes of Sets
Problems and Complements
2.14 Maximal Inequalities and Tail Bounds
2.14.1 Uniform Entropy
2.14.2 Bracketing Entropy
2.14.3 Mixed Entropy
2.14.4 VC-Classes
2.14.5 Bounds on Orlicz Norms
2.14.6 Tail Bounds
2.14.6. 1Proofs
2.14.6.2Proof of Theorem 2.14.32
Problems and Complements
2.15 Concentration
2.15.1 Bounded Difference Inequality
2.15.2 Talagrand’s Inequality
Problems and Complements
2 Notes
PART 3 Statistical Applications
3.1 Introduction
Problems and Complements
3.2 M-Estimators
3.2.1 Argmax Theorem
3.2.2 Rate of Convergence
3.2.3 Examples
3.2.4 Linearization
3.2.4.1 I.i.d. observations
Problems and Complements
3.3 Z-Estimators
3.3.1 I.i.d. observations
3.4 Sieved and Penalized M-Estimators
3.4.1 Sieved Maximum Contrast Estimators
3.4.2 Continuity Modulus
3.4.3 Penalized Maximum Contrast Estimators
3.4.3.1 Smoothing Parameter
3.4.3.2 Posterior Mode for Gaussian priors
3.4.4 Maximum Likelihood
3.4.5 Concave Parametrizations
3.4.6 Least-Squares Regression with Fixed Design
3.4.7 Least-Squares Regression with Random Design
3.4.8 Least-Absolute-Deviation Regression
3.4.9 Penalized Least-Squares Regression
3.4.10 Lasso
3.4.11 Classification
3.4.12 Support Vector Machines
Problems and Complements
3.5 Model Selection
3.5.1 General Result
3.5.2 Statistical Learning
Problems and Complements
3.6 Random Sample Size, Poissonization and Kac Processes
3.6.1 Random Sample Size
3.6.2 Poissonization
3.7 Bootstrap
3.7.1 Empirical Bootstrap
3.7.2 Exchangeable Bootstrap
Problems and Complements
3.8 Two-Sample Problem
3.8.1 Permutation Empirical Processes
3.8.2 Two-Sample Bootstrap
Problems and Complements
3.9 Independence Empirical Processes
Problems and Complements
3.10 Delta-Method
3.10.1 Main Result
3.10.2 Gaussian Limits
3.10.3 Conditional Delta-Method
3.10.4 Random Centering
3.10.5 Examples
3.10.5.1 Wilcoxon Statistic
3.10.5.2 Inverse Map
3.10.5.1 Composition
3.10.5.4 Copula Function
3.10.5.5 The Product Integral
3.10.5.6 Multivariate Trimming
3.10.5.7 Z-Functionals
Problems and Complements
3.11 Contiguity
3.11.1 Empirical Process
3.11.2 Change-Point Alternatives
Problems and Complements
3.12 Convolution and Minimax Theorems
3.12.1 Efficiency of the Empirical Distribution
Problems and Complements
3.13 Random Empirical Processes
3.13.1 Uniform Function Classes
3.13.2 Weakly Converging Estimators
3.13.3 Composition
3.13.3.1 Smooth Functions
3.13.3.2 Functions of Bounded Variation
3.13.4 Examples
3.13.4.1 Regression Residuals
3.13.4.2 Kendall’s Process
3.13.3 Copula Function
Problems and Complements
3 Notes
PART A Appendix
A.1 Inequalities
Problems and Complements
Problems and Complements
A.2 Gaussian Processes
A.2.1 Maximal Inequalities and Comparison
A.2.2 Geometric Inequalities
A.2.3 Exponential Bounds
A.2.4 Argmax
Problems and Complements
A.3 Rademacher Processes
A.4 Isoperimetric Inequalities for Product Measures
A.5 Some Limit Theorems
A.6 More Inequalities
A.6.1 Binomial Random Variables
A.6.2 Multinomial Random Vectors
A.6.3 Rademacher Sums
Problems and Complements
A Notes
References
Author Index
Subject Index
List of Symbols