Self-Normalized Processes: Limit Theory and Statistical Applications

byVictor H. Pe, Tze Leung Lai, Qi-Man Shao

Paperback | November 30, 2010

Self-Normalized Processes: Limit Theory and Statistical Applications by Victor H. Pe
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Self-normalized processes are of common occurrence in probabilistic and statistical studies. A prototypical example is Student''s t-statistic introduced in 1908 by Gosset, whose portrait is on the front cover. Due to the highly non-linear nature of these processes, the theory experienced a long period of slow development. In recent years there have been a number of important advances in the theory and applications of self-normalized processes. Some of these developments are closely linked to the study of central limit theorems, which imply that self-normalized processes are approximate pivots for statistical inference.

The present volume covers recent developments in the area, including self-normalized large and moderate deviations, and laws of the iterated logarithms for self-normalized martingales. This is the first book that systematically treats the theory and applications of self-normalization.

Victor H. de la Peña is Fellow of Institute of Mathematical Statistics and a Medallion Lecturer for IMS in 2007.Tze Leung LAI: Distinguished Lecture Series in Statistical Science from Academia Sinica (2001), Starr Lectures in Financial Mathematics from the University of Hong Kong (2001), Center for Advanced Study i...
Title:Self-Normalized Processes: Limit Theory and Statistical ApplicationsFormat:PaperbackProduct dimensions:275 pages, 9.25 X 6.1 X 0 inShipping dimensions:275 pages, 9.25 X 6.1 X 0 inPublished:November 30, 2010Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3642099262

ISBN - 13:9783642099267

Appropriate for ages: All ages

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Table of Contents

Independent Random Variables.- Classical Limit Theorems, Inequalities and Other Tools.- Self-Normalized Large Deviations.- Weak Convergence of Self-Normalized Sums.- Stein''s Method and Self-Normalized Berry-Esseen Inequality.- Self-Normalized Moderate Deviations and Laws of the Iterated Logarithm.- Cramér-Type Moderate Deviations for Self-Normalized Sums.- Self-Normalized Empirical Processes and U-Statistics.- Martingales and Dependent Random Vectors.- Martingale Inequalities and Related Tools.- A General Framework for Self-Normalization.- Pseudo-Maximization via Method of Mixtures.- Moment and Exponential Inequalities for Self-Normalized Processes.- Laws of the Iterated Logarithm for Self-Normalized Processes.- Multivariate Self-Normalized Processes with Matrix Normalization.- Statistical Applications.- The t-Statistic and Studentized Statistics.- Self-Normalization for Approximate Pivots in Bootstrapping.- Pseudo-Maximization in Likelihood and Bayesian Inference.- Sequential Analysis and Boundary Crossing Probabilities for Self-Normalized Statistics.

Editorial Reviews

From the reviews: "Readership: Research workers in applied probability. . it serves as a reference text for a special-topic course for PhD students; each chapter after the first ends with a collection of problems and the material is based on such a course taught by two of the authors at Stanford and Hong kong. . It is a thorough . study of an area of applied probability that underlies important statistical methodology. . I am sure that the text will encourage others to join them in their work." (Martin Crowder, International Statistical Review, Vol. 77 (3), 2009) "The monograph will certainly be of great use as a reference text for researchers working on corresponding problems, but also for Ph.D. and other advanced students who want to learn about the techniques and relevant topics in an interesting and active research area. . this monograph provides a very useful collection of recent and earlier research results in the theory and applications of self-normalized processes and can be used as a standard reference text by graduate students and researchers in the field." (Josef Steinebach, Zentralblatt MATH, Vol. 1165, 2009) "This book covers recent developments on self-normalized processes, emphasizing important advances in the area. It is the first book that systematically treats the theory and applications of self-normalized processes. . In all aspects, this is an excellent book, and it is ideal for a second-year Ph.D. level topics course. It is also a great book for anyone who is interested in research in self-normalized processes and related areas." (Fuchang Gao, Mathematical Reviews, Issue 2010 d)