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.
1 Basic Convergence Concepts and Theorems 1
2 Metrics, Information Theory, Convergence, and Poisson Approximations 19
3 More General Weak and Strong Laws and the Delta Theorem 35
4 Transformations 49
5 More General Central Limit Theorems 63
6 Moment Convergence and Uniform Integrability 83
7 Sample Percentiles and Order Statistics 91
8 Sample Extremes 101
9 Central Limit Theorems for Dependent Sequences 119
10 Central Limit Theorem for Markov Chains 131
11 Accuracy of Central Limit Theorems 141
12 Invariance Principles 151
13 Edgeworth Expansions and Cumulants 185
14 Saddlepoint Approximations 203
15 U-statistics 225
16 Maximum Likelihood Estimates 235
17 M Estimates 259
18 The Trimmed Mean 271
19 Multivariate Location Parameter and Multivariate Medians 279
20 Bayes Procedures and Posterior Distributions 289
21 Testing Problems 323
22 Asymptotic Efficiency in Testing 347
23 Some General Large-Deviation Results 365
24 Classical Nonparametrics 377
25 Two-Sample Problems 401
26 Goodness of Fit
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