Asymptotic Theory of Statistics and Probability

ISBN-13: 9781461498841 / Angielski / Miękka / 2014 / 722 str.

Anirban DasGupta

Opis

This book developed out of my year-long course on asymptotic theory at Purdue University. To some extent, the topics coincide with what I cover in that course. There are already a number of well-known books on asy- totics. This book is quite different. It covers more topics in one source than areavailableinanyothersinglebookonasymptotictheory. Numeroustopics covered in this book are available in the literature in a scattered manner, and they are brought together under one umbrella in this book. Asymptotic theory is a central unifying theme in probability and statistics. My main goal in writing this book is to give its readers a feel for the incredible scope and reach of asymptotics. I have tried to write this book in a way that is accessible and to make the reader appreciate the beauty of theory and the insights that only theory can provide. Essentially every theorem in the book comes with at least one reference, preceding or following the statement of the theorem. In addition, I have p- vided a separate theorem-by-theorem reference as an entry on its own in the front of the book to make it extremely convenient for the reader to ?nd a proof that was not provided in the text. Also particularly worth mentioning is a collection of nearly 300 practically useful inequalities that I have c- lected together from numerous sources. This is appended at the very end of the book.

Recenzje

From the reviews:

"This is definitely not your typical book on theory. The approach that Dasgupta has taken with this far-reaching volume is to explore important results and applications of asymptotic theory without emphasizing the intricate mathematical details. The focus is on the forest rather than on the trees, and this results in a readable text that, for the most part, should be accessible to anyone with a first-year graduate-level course in statistical theory.

I would imagine that this book woult be very useful as a first place to look for help in solving many problems in asymptotics. The book can provide an overview of the key issues, some ideas, and a path to more detail." (Biometrics, September 2008)

"Presents an encyclopedic treatment of classic as well as contemporary large sample theory, including both statistical problems and probabilistic issues and tools." (Journal of Economic Literature, Volume 46, no. 3, 2008)

"... a nice handbook and reference material. This is a different book on the asymptotic theory and its use in probability and statistical inference. It covers a wide range of divergent topics where the large sample theory is useful and can be naturally applied. ... The book is will organized and clearly written. The book works well as a reference text for a theoretical statistician working with the asymptotics. It can also be used as a textbook for several topics of the graduate courses." (International Statistical Review,2009, 77, 1)

"This book provides a comprehensive overview of asymptotic theory in probability and mathematical statistics. ... The text is written in a very clear style ... . the book is a very good choice as a first reading. It also contains a large collection of inequalities from linear algebra, probability and analysis that are of importance in mathematical statistics. This collection makes the volume even more valuable as a reference for students as well as research workers." (Zentralblatt MATH, Vol. 1154, 2009)

"This amazing book covers an enormous variety of topics from modern statistics and probability...Among the great achievements of the author is not only the panoramic coverage of modern stochastics but also in demonstrating convincingly the fundamental role of probability theory in any kind of statistical inference problems...An unreplaceable source of information for anyone studying probability and statistics...An invaluable reference to be acquired by any good science library." (Journal of the Royal Statistical Society)

"Contains a total of 35 (!) chapters, covering both theoretical foundations and many ... applications of asymptotic statistics. The chapters can broadly be classified in two categories: more classic large sample theory chapters, and chapters containing topics which are usually not treated in other books on asymptotics. ... I would recommend this book to readers who have attended courses on probability theory ... and mathematical statistics. ... Someone who searches a good and exhaustive reference book for asymptotic statistics ... will certainly appreciate this book." (Björn Bornkamp, Statistical Papers, Vol. 51, 2010)

"This book provides a very broad coverage of both classical and contemporary topics, with an emphasis on the conceptual discussion of results, issues, tools and implications. ... This makes the book quite different from other books on asymptotics and provides an invaluable reference for anyone studying probability and statistics. It can be used to design graduate-level courses with various emphases, to assign for independent reading, and to have a comprehensive overview of asymptotic theory." (Wenbo V. Li, Mathematical Reviews, Issue 2011 m)

Spis treści

Basic Convergence Concepts and Theorems.- Metrics, Information Theory, Convergence, and Poisson Approximations.- More General Weak and Strong Laws and the Delta Theorem.- Transformations.- More General Central Limit Theorems.- Moment Convergence and Uniform Integrability.- Sample Percentiles and Order Statistics.- Sample Extremes.- Central Limit Theorems for Dependent Sequences.- Central Limit Theorem for Markov Chains.- Accuracy of Central Limit Theorems.- Invariance Principles.- Edgeworth Expansions and Cumulants.- Saddlepoint Approximations.- U-statistics.- Maximum Likelihood Estimates.- M Estimates.- The Trimmed Mean.- Multivariate Location Parameter and Multivariate Medians.- Bayes Procedures and Posterior Distributions.- Testing Problems.- Asymptotic Efficiency in Testing.- Some General Large-Deviation Results.- Classical Nonparametrics.- Two-Sample Problems.- Goodness of Fit.- Chi-square Tests for Goodness of Fit.- Goodness of Fit with Estimated Parameters.- The Bootstrap.- Jackknife.- Permutation Tests.- Density Estimation.- Mixture Models and Nonparametric Deconvolution.- High-Dimensional Inference and False Discovery.- A Collection of Inequalities in Probability, Linear Algebra, and Analysis.

Blurb

This book is an encyclopedic treatment of classic as well as contemporary large sample theory, dealing with both statistical problems and probabilistic issues and tools. It is written in an extremely lucid style, with an emphasis on the conceptual discussion of the importance of a problem and the impact and relevance of the theorems. The book has 34 chapters over a wide range of topics, nearly 600 exercises for practice and instruction, and another 300 worked out examples. It also includes a large compendium of 300 useful inequalities on probability, linear algebra, and analysis that are collected together from numerous sources, as an invaluable reference for researchers in statistics, probability, and mathematics.

It can be used as a graduate text, as a versatile research reference, as a source for independent reading on a wide assembly of topics, and as a window to learning the latest developments in contemporary topics. The book is unique in its detailed coverage of fundamental topics such as central limit theorems in numerous setups, likelihood based methods, goodness of fit, higher order asymptotics, as well as of the most modern topics such as the bootstrap, dependent data, Bayesian asymptotics, nonparametric density estimation, mixture models, and multiple testing and false discovery. It provides extensive bibliographic references on all topics that include very recent publications.

Anirban DasGupta is Professor of Statistics at Purdue University. He has also taught at the Wharton School of the University of Pennsylvania, at Cornell University, and at the University of California at San Diego. He has been on the editorial board of the Annals of Statistics since 1998 and has also served on the editorial boards of the Journal of the American Statistical Association, International Statistical Review, and the Journal of Statistical Planning and Inference. He has edited two monographs in the lecture notes monograph series of the Institute of Mathematical Statistics, is a Fellow of the Institute of Mathematical Statistics and has 70 refereed publications on theoretical statistics and probability in major journals.