Probability for Statisticians

ISBN-13: 9783319522067 / Angielski / Miękka / 2017 / 510 str.

Galen Shorack

Opis

Probability for Statisticians is intended as a text for a one year graduate course aimed especially at students in statistics. The choice of examples illustrates this intention clearly. The material to be presented in the classroom constitutes a bit more than half the text, and the choices the author makes at the University of Washington in Seattle are spelled out. The rest of the text provides background, offers different routes that could be pursued in the classroom, ad offers additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic funcion presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. The martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage. The author is a professor of Statistics and adjunct professor of Mathematics at the University of Washington in Seattle. He served as chair of the Department of Statistics 1986-- 1989. He received his PhD in Statistics from Stanford University. He is a fellow of the Institute of Mathematical Statistics, and is a former associate editor of the Annals of Statistics.

Recenzje

"It discusses measure theoretic probability from the viewpoint of what a theoretical statistician needs to know, and includes many details that an applied statistician may need to look up on occasion. Reading it frequently feels like you are sitting next to the author, with him pointing out the important parts, and suggesting how to think about things. I enjoyed that aspect very much, and it helps to solidify the readers understanding." (Peter Rabinovitch, MAA Reviews, January, 2018)

Spis treści

Preface

Use of This Text

Definition of Symbols

Chapter 1. Measures

1. Basic Properties of Measures
2. Construction and Extension of Measures
3. Lebesgue Stieltjes Measures

Chapter 2. Measurable Functions and Convergence

1. Mappings and σ-Fields
2. Measurable Functions
3. Convergence
4. Probability, RVs, and Convergence in Law
5. Discussion of Sub σ-Fields

Chapter 3. Integration

1. The Lebesgue Integral
2. Fundamental Properties of Integrals
3. Evaluating and Differentiating Integrals
4. Inequalities
5. Modes of Convergence

Chapter 4 Derivatives via Signed Measures

1. Introduction
2. Decomposition of Signed Measures
3. The Radon Nikodym Theorem
4. Lebesgue's Theorem
5. The Fundamental Theorem of Calculus

Chapter 5. Measures and Processes on Products

1. Finite-Dimensional Product Spaces
2. Random Vectors on (Ω,Α,P)
3. Countably Infinite Product Probability Spaces
4. Random Elements and Processes on (Ω,Α,P)

Chapter 6. Distribution and Quantile Functions

1. Character of Distribution Functions
2. Properties of Distribution Functions
3. The Quantile Transformation
4. Integration by Parts Applied to Moments
5. Important Statistical Quantities
6. Infinite Variances

Chapter 7. Independence and Conditional Distributions

1. Independence
2. The Tail σ-Field
3. Uncorrelated Random Variables
4. Basic Properties of Conditional Expectation
5. Regular Conditional Probability

Chapter 8. WLLN, SLLN, LIL, and Series

1. Introduction
2. Borel Cantelli and Kronecker Lemmas
3. Truncation, WLLN, and Review of Inequalities
4. Maximal Inequalities and Symmetrization
5. The Classical Laws of Large Numbers (or, LLNs)
6. Applications of the Laws of Large Numbers
7. Law of the Iterated Logarithm (or, LIL)
8. Strong Markov Property for Sums of IID RVs
9. Convergence of Series of Independent RVs
10. Martinagles
11. Maximal Inequalities, Some with ↗ Boundaries

Chapter 9. Characteristic Functions and Determining Classes

1. Classical Convergence in Distribution
2. Determining Classes of Functions
3. Characteristic Functions, with Basic Results
4. Uniqueness and Inversion
5. The Continuity Theorem
6. Elementary Complex and Fourier Analysis
7. Esseen's Lemma
8. Distributions on Grids
9. Conditions for Ø to Be a Characteristic Function

Chapter 10. CLTs via Characteristic Functions

1. Introduction
2. Basic Limit Theorems
3. Variations on the Classical CLT
4. Examples of Limiting Distributions
5. Local Limit Theorems
6. Normality Via Winsorization and Truncation
7. Identically Distributed RVs
8. A Converse of the Classical CLT
9. Bootstrapping
10. Bootstrapping with Slowly ↗ Winsorization

Chapter 11. Infinitely Divisible and Stable Distributions

1. Infinitely Divisible Distributions
2. Stable Distributions
3. Characterizing Stable Laws
4. The Domain of Attraction of a Stable Law
5. Gamma Approximations
6. Edgeworth Expansions

Chapter 12. Brownian Motion and Empirical Processes

1. Special Spaces 
2. Existence of Processes on (C, C) and (D, D)
3. Brownian Motion and Brownian Bridge
4. Stopping Times
5. Strong Markov Property
6. Embedding a RV in Brownian Motion
7. Barrier Crossing Probabilities
8. Embedding the Partial Sum Process
9. Other Properties of Brownian Motion
10. Various Empirical Processes
11. Inequalities for the Various Empirical Processes
12. Applications

Chapter 13. Martingales

1. Basic Technicalities for Martingales
2. Simple Optional Sampling Theorem
3. The Submartingale Convergence Theorem
4. Applications of the S-mg Convergence Theorem
5. Decomposition of a Submartingale Sequence
6. Optional Sampling
7. Applications of Optional Sampling
8. Introduction to Counting Process Martingales
9. CLTs for Dependent RVs

Chapter 14. Convergence in Law on Metric Spaces

1. Convergence in Distribution on Metric Spaces
2. Metrics for Convergence in Distribution 

Chapter 15. Asymptotics Via Empirical Processes

1. Introduction
2. Trimmed and Winsorized Means
3. Linear Rank Statistics and Finite Sampling
4. L-Statistics

Appendix A. Special Distributions

Elementary Probability
1. Distribution Theory for Statistics
Appendix B. General Topology and Hilbert Space
1. General Topology
2. Metric Spaces
3. Hilbert Space
Appendix C. More WLLN and CLT
1. Introduction
2. General Moment Estimation Specific
3. Slowly Varying Partial Variance when σ2=∞
4. Specific Tail Relationships
5. Regularly Varying Functions
6. Some Winsorized Variance Comparison
7. Inequalities for Winsorized Quantile Functions
References
Index

O autorze

Galen Shorack, PhD, is Professor Emeritus in the Department of Statistics (of which he was a founding member) and Adjunct Professor in the Department of Mathematics at the University of Washington, Seattle.  He received his Bachelor of Science and Master of Science degrees in Mathematics from the University of Oregon and his PhD in Statistics from Stanford University.  Dr. Shorack's research interests include limit theorems in statistics, the theory of empirical processes, trimming-Winsorizing, and regular variation.  He has served as Associate Editor of the Annals of Mathematical Statistics (Annals of Statistics) and is Fellow of the Institute of Mathematical Statistics.

Blurb

This 2nd edition textbook offers a rigorous introduction to measure theoretic probability with particular attention to topics of interest to mathematical statisticians—a textbook for courses in probability for students in mathematical statistics. It is recommended to anyone interested in the probability underlying modern statistics, providing a solid grounding in the probabilistic tools and techniques necessary to do theoretical research in statistics. For the teaching of probability theory to post graduate statistics students, this is one of the most attractive books available.

Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. Martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage. 

This is a heavily reworked and considerably shortened version of the first edition of this textbook. "Extra" and background material has been either removed or moved to the appendices and important rearrangement of chapters has taken place to facilitate this book's intended use as a textbook.

New to this edition:

• Still up front and central in the book, Chapters 1-5 provide the "measure theory" necessary for the rest of the textbook and Chapters 6-7 adapt that measure-theoretic background to the special needs of probability theory
• Develops both mathematical tools and specialized probabilistic tools
• Chapters organized by number of lectures to cover requisite topics, optional lectures, and self-study
• Exercises interspersed within the text
• Guidance provided to instructors to help in choosing topics of emphasis