Empirical Processes with Applications to Statistics (Classics in Applied Mathematics, Series Number 59)

✍ Scribed by Galen R. Shorack, Jon A. Wellner

• Publisher: Society for Industrial and Applied Mathematics
• Year: 2009
• Tongue: English
• Leaves: 1000
• Edition: Revised ed.
• Category: Library

✦ Synopsis

Originally published in 1986, this valuable reference provides a detailed treatment of limit theorems and inequalities for empirical processes of real-valued random variables. It also includes applications of the theory to censored data, spacings, rank statistics, quantiles, and many functionals of empirical processes, including a treatment of bootstrap methods, and a summary of inequalities that are useful for proving limit theorems. At the end of the Errata section, the authors have supplied references to solutions for 11 of the 19 Open Questions provided in the book's original edition.

✦ Table of Contents

Empirical Processes with Applications to Statistics
Short Table of Contents
Contents
List of Tables
Preface to the Classics Edition
The Uniform Song
Preface
Acknowledgments and Sources of Tables
List of Special Symbols
1 Introduction and Survey of Results
1. Definition of the Empirical Process and the Inverse Transformation
2. Survey of Results for .U..
3. Results for the Random Functions G. and U. on [0, 1]
4. Convergence of U. in Other Metrics
5. Survey of Other Results
2 Foundations, Special Spaces and Special Processes
0. Introduction
1. Random Elements, Processes, and Special Spaces
2. Brownian Motions S, Brownian Bridge U, the Uhlenbeck Process, the Kiefer Process K, the Brillinger Process
3. Weak Convergence .
4. Weak Convergence of the Partial-Sum Process S.
5. The Skorokhod Embedding of Partial Sums
6. Wasserstein Distance
7. The Hungarian Construction of Partial Sums
8. Relative Compactness .
9. Relative Compactness of S(nI)/vn.b.
10. Weak Convergence of the Maximum of Normalized Brownian Motion and Partial Sums
11. The LLN for iid rv's
3 Convergence and Distributions of Empirical Processes
1. Uniform Processes and Their Special Construction
2. Definition of Some Basic Processes under General Alternatives
3. Weak Convergence of the General Weighted Empirical Process
4. The Special Construction for a Fixed Nearly Null Array
5. The Sequential Uniform Empirical Process K.
6. Martingales Associated with U., V., W., and R.
7. A Simple Result on Convergence in Metrics
8. Limiting Distributions under the Null Hypothesis
4 Alternatives and Processes of Residuals
0. Introduction
1. Contiguity
2. Limiting Distributions under Local Alternatives
3. Asymptotic Optimality of F.
4. Limiting Distributions under Fixed Alternatives
5. Convergence of Empirical and Rank Processes under Contiguous Location, Scale, and Regression Alternatives
6. Empirical and Rank Processes of Residuals
5 Integral Tests of Fit and Estimated Empirical Process
0. Introduction
1. Motivation of Principal Component Decomposition
2. Orthogonal Decomposition of Processes
3. Principal Component Decomposition of U., U and Other Related Processes
4. Principal Component Decomposition of the Anderson and Darling Statistic A..
5. Tests of Fit with Parameters Estimated
6. The Distribution of W., W.., A., A.., and Other Related Statistics
7. Confidence Bands, Acceptance Bands, and QQ, PP, and SP Plots
8. More on Components
9. The Minimum Cramér-von Mises Estimate of Location
6 Martingale Methods
0. Introduction
1. The Basic Martingale M. for U.
2. Processes of the Form .M., .U.(F), and .W.(F)
3. Processes of the Form . h dM.
4. Processes of the Form . h dU.(F) and . h dW.(F)
5. Processes of the Form . M. dh, . U.(F) dh, and . W.(F) dh
6. Reductions When F is Uniform
7 Censored Data and the Product-Limit Estimator
0. Introduction
1. Convergence of the Basic Martingale M.
2. Identities Based on Integration by Parts
3. Consistency of ... and F..
4. Preliminary Weak Convergence . of B. and X.
5. Martingale Representations
6. Inequalities
7. Weak Convergence . of B. and X. in ..-Metrics
8. Extension to General Censoring Times
8 Poisson and Exponential Representations
0. Introduction
1. The Poisson Process N
2. Representations of Uniform Order Statistics
3. Representations of Uniform Quantile Processes
4. Poisson Representations of U.
5. Poisson Embeddings
9 Some Exact Distributions
0. Introduction
1. Evaluating the Probability that G. Crosses a General Line
2. The Exact Distribution of .U.±. and the DKW Inequality
3. Recursions for P(g . G. . h)
4. Some Combinatorial Lemmas
5. The Number of Intersections of G. with a General Line
6. On the Location of the Maximum of U.. and U..
7. Dwass's Approach to G. Based on Poisson Processes
8. Local Time of U.
9. The Two-Sample Problem
10 Linear and Nearly Linear Bounds on the Empirical Distribution Function G.
0. Summary
1. Almost Sure Behavior of .n:k with k Fixed
2. A Glivenko-Cantelli-type Theorem for .(G.-I)..
3. Inequalities for the Distributions of ¦G./I¦ and ¦I/G.¦..n:1
4. In-Probability Linear Bounds on G.
5. Characterization of Upper-Class Sequences for ¦G./I¦ and ¦I/G.¦..n:1
6. Almost Sure Nearly Linear Bounds on G. and G.
7. Bounds on Functions of Order Statistics
8. Almost Sure Behavior of Z.(a.)/b. as a.v0
9. Almost Sure Behavior of Normalized Quantiles as a.v0
11 Exponential Inequalities and ¦•/q¦-Metric Convergence of U. and V.
0. Introduction
1. Universal Exponential Bounds for Binomial rv's
2. Bounds on the Magnitude of ¦U.../q¦..
3. Exponential Bounds for Uniform Order Statistics
4. Bounds for the Magnitude of ¦V.../q¦..
5. Weak Convergence of U. and V. in ¦•/q¦ Metrics
6. Convergence of U., W., .. and R.. in Weighted .. Metrics
7. Moments of Functions of Order Statistics
8. Additional Binomial Results
9. Exponential Bounds for Poisson, Gamma, and Beta rv's
12 The Hungarian Constructions of K., U. and V.
0. Introduction
1. The Hungarian Construction of K.
2. The Hungarian Renewal Construction of ..
3. A Refined Construction of U. and V.
4. Rate of Convergence of the Distribution of Functionals
13 Laws of the Iterated Logarithm Associated with U. and V.
0. Introduction
1. A LIL for ¦U.#¦
2. A Maximal Inequality for ..
3. Relative Compactness . of U. and V.
4. Relative Compactness of U. in -Metrics
5. The Other LIL for ¦U.¦
6. Extension to General F
14 Oscillations of the Empirical Process
0. Introduction
1. The Oscillation Moduli ., .., and . of U and S.
2. The Oscillation Moduli of U.
3. A Modulus of Continuity for the Kiefer Process K.
4. The Modulus of Continuity Again, via the Hungarian Construction
5. Exponential Inequalities for Poisson Processes
6. The Modulus of Continuity Again, via Poisson Embedding
7. The Modulus of Continuity of V.
15 The Uniform Empirical Difference Process D. . U. +V.
0. Introduction
1. The Uniform Empirical Difference Process D.
2. The Integrated Empirical Difference Process
16 The Normalized Uniform Empirical Process Z. and the Normalized Uniform Quantile Process
0. Introduction
1. Weak Convergence of ¦Z.¦
2. The a.s. Rate of Divergence of .
3. Almost Sure Behavior of with a..0
4. The a.s. Divergence of the Normalized Quantile Process
17 The Uniform Empirical Process Indexed by Intervals and Functions
0. Introduction
1. Bounds on the Magnitude of ¦ .(a,b)
2. Weak Convergence of U. in ¦ Metrics
3. Indexing by Continuous Functions via Chaining
18 The Standardized Quantile Process Q.
0. Introduction
1. Weak Convergence of the Standardized Quantile Process Q.
2. Approximation of Q. by V. with Applications
3. Asymptotic Theory of the Q-Q Plot
4. Weak Convergence . of the Product-Limit Quantile Process Y.
19 L-Statistics
0. Introduction
1. Statement of the Theorems
2. Some Examples of L-statistics
3. Randomly Trimmed and Winsorized Means
4. Proofs
20 Rank Statistics
0. Linear Rank Statistics
1. The Basic Martingale M.
2. Processes of the Form T. = . h. dR. in the Null Case
3. Contiguous Alternatives
4. The Chernoff and Savage Theorem
5. Some Exercises for Order Statistics and Spacings
21 Spacings
0. Introduction
1. Definitions and Distributions of Uniform Spacings
2. Limiting Distributions of Ordered Uniform Spacings
3. Renewal Spacings Processes
4. Uniform Spacings Processes
5. Testing Uniformity with Functions of Spacings
6. Iterated Logarithms for Spacings
22 Symmetry
1. The Empirical Symmetry Process S. and the Empirical Rank Symmetry Process R.
2. Testing Goodness of Fit for a Symmetric DF
3. The Processes under Contiguity
4. Signed Rank Statistics under Symmetry
5. Estimating an Unknown Point of Symmetry
6. Estimating the DF of a Symmetric Distribution with Unknown Point of Symmetry
23 Further Applications
1. Bootstrapping the Empirical Process
2. Smooth Estimates of F
3. The Shorth
4. Convergence of U-Statistic Empirical Processes
5. Reliability and Econometric Functions
24 Large Deviations
0. Introduction
1. Bahadur Efficiency
2. Large Deviations for Supremum Tests of Fit
3. The Kullback-Leibler Information Number
4. The Sanov Problem
25 Independent but not Identically Distributed Random Variables
0. Introduction
1. Extensions of the DKW Inequality
2. The Generalized Binomial Distribution
3. Bounds on F.
4. Convergence of X., Y., and Z. with respect to Metrics
5. More on L-statistics
26 Empirical Measures and Processes for General Spaces
0. Introduction
1. Glivenko-Cantelli Theorems via the Vapnik-Chervonenkis Idea
2. Glivenko-Cantelli Theorems via Metric Entropy
3. Weak and Strong Approximations to the Empirical Process Z.
Appendix A: Inequalities and Miscellaneous
0. Introduction
1. Simple Moment Inequalities
2. Maximal Inequalities for Sums and a Minimal Inequality
3. Berry-Esseen Inequalities
4. Exponential Inequalities and Large Deviations
5. Moments of Sums
6. Borel-Cantelli Lemmas
7. Miscellaneous Inequalities
8. Miscellaneous Probabilistic Results
9. Miscellaneous Deterministic Results
10. Martingale Inequalities
11. Inequalities for Reversed Martingales
12. Inequalities in Higher Dimensions
13. Finite-Sampling Inequalities
14. Inequalities for Processes
Appendix B: Martingales and Counting Processes
1. Basic Terminology and Definitions
2. Counting Processes and Martingales
3. Stochastic Integrals for Counting Processes
4. Martingale Inequalities
5. Rebolledo's Martingale Central Limit Theorem
6. A Change of Variable Formula and Exponential Semimartingales
Errata
1 Introduction
2 Major changes and revisions
2.1 Revision and correction of section 7.3
2.2 Revision and correction of Section 7.7
2.3 Revision and correction of Section 19.4
2.4 Revision and correction of Section 23.3
3 Typographical errors, spelling errors, and minor changes
4 Accent mark revisions
5 Solutions of Open Questions
References
References
Author Index
Subject Index