Asymptotic theory of statistics and probability

Anirban DasGupta (auth.)0387759700, 9780387759708

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.

Table of contents :
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