Asymptotic Theory of Statistics and Probability (Springer Texts in Statistics)

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 Revi__ew, 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. 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 Revi ew, 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. Front Matter....Pages I-XXVII Basic Convergence Concepts and Theorems....Pages 1-17 Metrics, Information Theory, Convergence, and Poisson Approximations....Pages 19-34 More General Weak and Strong Laws and the Delta Theorem....Pages 35-47 Transformations....Pages 49-61 More General Central Limit Theorems....Pages 63-81 Moment Convergence and Uniform Integrability....Pages 83-89 Sample Percentiles and Order Statistics....Pages 91-100 Sample Extremes....Pages 101-117 Central Limit Theorems for Dependent Sequences....Pages 119-129 Central Limit Theorem for Markov Chains....Pages 131-140 Accuracy of Central Limit Theorems....Pages 141-149 Invariance Principles....Pages 151-183 Edgeworth Expansions and Cumulants....Pages 185-201 Saddlepoint Approximations....Pages 203-224 U -statistics....Pages 225-234 Maximum Likelihood Estimates....Pages 235-258 M Estimates....Pages 259-269 The Trimmed Mean....Pages 271-278 Multivariate Location Parameter and Multivariate Medians....Pages 279-288 Bayes Procedures and Posterior Distributions....Pages 289-321 Testing Problems....Pages 323-345 Asymptotic Efficiency in Testing....Pages 347-364 Some General Large-Deviation Results....Pages 365-376 Classical Nonparametrics....Pages 377-399 Two-Sample Problems....Pages 401-419 Goodness of Fit....Pages 421-439 Chi-square Tests for Goodness of Fit....Pages 441-450 Goodness of Fit with Estimated Parameters....Pages 451-459 The Bootstrap....Pages 461-497 Jackknife....Pages 499-512 Permutation Tests....Pages 513-521 Density Estimation....Pages 523-570 Mixture Models and Nonparametric Deconvolution....Pages 571-591 High-Dimensional Inference and False Discovery....Pages 593-631 A Collection of Inequalities in Probability, Linear Algebra, and Analysis....Pages 633-687 Back Matter....Pages 689-722 "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."--Jacket 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.

This unique book delivers an encyclopedic treatment of classic as well as contemporary large sample theory, dealing with both statistical problems and probabilistic issues and tools. The book is unique in its detailed coverage of fundamental topics. 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. There is no other book in large sample theory that matches this book in coverage, exercises and examples, bibliography, and lucid conceptual discussion of issues and theorems.