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Description
Although Normal Approximation and Asymptotic Expansions was first published in 1976, it has gained new significance and renewed interest among statisticians due to the developments of modern statistical techniques such as the bootstrap, the efficacy of which can be ascertained by asymptotic expansions.
This also is the only book containing a detailed treatment of various refinements of the multivariate central limit theorem (CLT), including Berry-Essen-type error bounds for probabilities of general classes of functions and sets, and asymptotic expansions for both lattice and non-lattice distributions. With meticulous care, the authors develop necessary background on
• weak convergence theory,
• Fourier analysis,
• geometry of convex sets, and
• the relationship between lattice random vectors and discrete subgroups of Rk.
The formalism developed in the book has been used in the extension of the theory by Goetze and Hipp to sums of weakly dependent random vectors.
This edition of the book includes a new chapter that provides an application of Stein's method of approximation to the multivariate CLT.
Keywords: Berry-Essen bounds, Asymptotic Expansions
Table of Contents
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Front Matter
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6. Two Recent Improvements
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Back Matter
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Excerpt
This monograph presents in a unified way various refinements of the classical central limit theorem for independent random vectors and includes recent research on the subject. Most of the multidimensional results in this area are fairly recent, and significant advances over the last 15 years have led to a fresh outlook. The increasing demands of application (e.g., to the large sample theory of statistics) indicate that the present generality is useful. It is rather fortunate that in our context precision and generality go hand in hand.
Apart from some material that most students in probability and statistics encounter during the first year of their graduate studies, this book is essentially self-contained. It is unavoidable that lengthy computations frequently appear in the text. We hope that in addition to making it easier for someone to check the veracity of a particular result of interest, the detailed computations will also be helpful in estimations of constants that appear in various error bounds in the text. To facilitate comprehension each chapter begins with a brief indication of the nature of the problem treated and its solution. Notes at the end of each chapter provide some history and references and, occasionally, additional facts. There is also an Appendix devoted partly to some elementary notions in probability and partly to some auxiliary results used in the book.
We have not discussed many topics closely related to the subject matter (not to mention applications). Some of these topics are “large deviation,” extension of the results of this monograph to the dependence case, and rates of convergence for the invariance principle. It would take another book of comparable size to cover these topics adequately.
©2010 SIAM
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