Asymptotic Theory of Statistics and Probability

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 im [...]
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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. TOC: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 Clts.- 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 Clts.- 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.