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Normal Approximation by Stein’s Method

  • Textbook
  • © 2011

Overview

Authors:
  1. Louis H.Y. Chen

    1. , Mathematics Department, National University of Singapore, Singapore, Singapore

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  2. Larry Goldstein

    1. , Department of Mathematics KAP 108, University of Southern California, Los Angeles, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Qi-Man Shao

    1. , Department of Mathematics and Technology, Hong Kong University of Science, Hong Kong, Hong Kong/PR China

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • First book that presents a complete self-contained treatment of Stein's method
  • Includes many recent applications
  • Important reference for researchers and suitable for graduate student seminars
  • Includes supplementary material: sn.pub/extras

Part of the book series: Probability and Its Applications (PIA)

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Table of contents (14 chapters)

  1. Front Matter

    Pages I-XII
    Download chapter PDF

  2. Introduction

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 1-12

  3. Fundamentals of Stein’s Method

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 13-44

  4. Berry–Esseen Bounds for Independent Random Variables

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 45-62

  5. L 1 Bounds

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 63-145

  6. L ∞ by Bounded Couplings

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 147-166

  7. L ∞: Applications

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 167-220

  8. Discretized Normal Approximation

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 221-232

  9. Non-uniform Bounds for Independent Random Variables

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 233-244

  10. Uniform and Non-uniform Bounds Under Local Dependence

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 245-255

  11. Uniform and Non-uniform Bounds for Non-linear Statistics

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 257-291

  12. Moderate Deviations

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 293-312

  13. Multivariate Normal Approximation

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 313-341

  14. Non-normal Approximation

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 343-369

  15. Group Characters and Malliavin Calculus

    • Louis H. Y. Chen, Larry Goldstein, Qi-Man Shao
    Pages 371-388

  16. Back Matter

    Pages 389-405
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Keywords

About this book

Since its introduction in 1972, Stein’s method has offered a completely novel way of evaluating the quality of normal approximations. Through its characterizing equation approach, it is able to provide approximation error bounds in a wide variety of situations, even in the presence of complicated dependence. Use of the method thus opens the door to the analysis of random phenomena arising in areas including statistics, physics, and molecular biology. Though Stein's method for normal approximation is now mature, the literature has so far lacked a complete self contained treatment. This volume contains thorough coverage of the method’s fundamentals, includes a large number of recent developments in both theory and applications, and will help accelerate the appreciation, understanding, and use of Stein's method by providing the reader with the tools needed to apply it in new situations. It addresses researchers as well as graduate students in Probability, Statistics and Combinatorics.

Reviews

From the reviews:

“This book gives a complete and comprehensive overview of normal approximation using Stein’s method. It serves both as an introductory textbook to graduate students who aim at becoming more familiar with Stein’s method and as a research book taking into account the most recent advances in this area. … To conclude, the authors present in this book the foundations and the main developments of Stein’s method … . The book is very pleasant to read and suitable for both graduate students and experienced researchers.” (Anthony Réveillac, Mathematical Reviews, Issue 2012 b)

Authors and Affiliations

  • , Mathematics Department, National University of Singapore, Singapore, Singapore

    Louis H.Y. Chen

  • , Department of Mathematics KAP 108, University of Southern California, Los Angeles, USA

    Larry Goldstein

  • , Department of Mathematics and Technology, Hong Kong University of Science, Hong Kong, Hong Kong/PR China

    Qi-Man Shao

About the authors

Louis Chen’s research interests are in probability and computational biology, focusing largely on Stein’s method. He is well-known for his pioneering work on Poisson approximation. He is an elected Fellow of the Institute of Mathematical Statistics and of the Academy of Sciences for the Developing World. He has also served as Associate Editor of Statistica Sinica and Bernoulli.

Larry Goldstein has studied Stein’s method since 1989, and is a noted researcher in the field. He was elected Fellow of the Institute of Mathematical Statistics in 2003, and serves on the editorial board of Bernoulli.

Qi-Man Shao has been working on limit theory in probability and statistics, especially on self-normalized large and moderate deviations and Stein’s method for normal and non-normal approximation. He is an invited speaker (45 minutes) at the International Congress of Mathematicians 2010. He is an elected Fellow of the Institute of Mathematical Statistics, and has served on the editorial board of The Annals of Statistics and The Annals of Applied Probability.

Bibliographic Information

  • Book Title: Normal Approximation by Stein’s Method

  • Authors: Louis H.Y. Chen, Larry Goldstein, Qi-Man Shao

  • Series Title: Probability and Its Applications

  • DOI: https://doi.org/10.1007/978-3-642-15007-4

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag GmbH Berlin Heidelberg 2011

  • Hardcover ISBN: 978-3-642-15006-7Published: 27 October 2010

  • Softcover ISBN: 978-3-642-26565-5Published: 01 December 2012

  • eBook ISBN: 978-3-642-15007-4Published: 13 October 2010

  • Series ISSN: 1431-7028

  • Edition Number: 1

  • Number of Pages: XII, 408

  • Number of Illustrations: 3 b/w illustrations

  • Topics: Probability Theory and Stochastic Processes

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