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Author: John E. Kolassa
Publisher: Springer Science & Business Media
ISBN: 1475742754
Category : Mathematics
Languages : en
Pages : 162
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Book Description
This book was originally compiled for a course I taught at the University of Rochester in the fall of 1991, and is intended to give advanced graduate students in statistics an introduction to Edgeworth and saddlepoint approximations, and related techniques. Many other authors have also written monographs on this subject, and so this work is narrowly focused on two areas not recently discussed in theoretical text books. These areas are, first, a rigorous consideration of Edgeworth and saddlepoint expansion limit theorems, and second, a survey of the more recent developments in the field. In presenting expansion limit theorems I have drawn heavily 011 notation of McCullagh (1987) and on the theorems presented by Feller (1971) on Edgeworth expansions. For saddlepoint notation and results I relied most heavily on the many papers of Daniels, and a review paper by Reid (1988). Throughout this book I have tried to maintain consistent notation and to present theorems in such a way as to make a few theoretical results useful in as many contexts as possible. This was not only in order to present as many results with as few proofs as possible, but more importantly to show the interconnections between the various facets of asymptotic theory. Special attention is paid to regularity conditions. The reasons they are needed and the parts they play in the proofs are both highlighted.
Author: John E. Kolassa
Publisher: Springer Science & Business Media
ISBN: 1475742754
Category : Mathematics
Languages : en
Pages : 162
Get Book Here
Book Description
This book was originally compiled for a course I taught at the University of Rochester in the fall of 1991, and is intended to give advanced graduate students in statistics an introduction to Edgeworth and saddlepoint approximations, and related techniques. Many other authors have also written monographs on this subject, and so this work is narrowly focused on two areas not recently discussed in theoretical text books. These areas are, first, a rigorous consideration of Edgeworth and saddlepoint expansion limit theorems, and second, a survey of the more recent developments in the field. In presenting expansion limit theorems I have drawn heavily 011 notation of McCullagh (1987) and on the theorems presented by Feller (1971) on Edgeworth expansions. For saddlepoint notation and results I relied most heavily on the many papers of Daniels, and a review paper by Reid (1988). Throughout this book I have tried to maintain consistent notation and to present theorems in such a way as to make a few theoretical results useful in as many contexts as possible. This was not only in order to present as many results with as few proofs as possible, but more importantly to show the interconnections between the various facets of asymptotic theory. Special attention is paid to regularity conditions. The reasons they are needed and the parts they play in the proofs are both highlighted.
Author: Springer
Publisher:
ISBN: 9781475742787
Category :
Languages : en
Pages : 204
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Book Description
Author: Martin A. Tanner
Publisher: Springer Science & Business Media
ISBN: 1468401920
Category : Mathematics
Languages : en
Pages : 166
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Book Description
This book provides a unified introduction to a variety of computational algorithms for likelihood and Bayesian inference. In this second edition, I have attempted to expand the treatment of many of the techniques dis cussed, as well as include important topics such as the Metropolis algorithm and methods for assessing the convergence of a Markov chain algorithm. Prerequisites for this book include an understanding of mathematical statistics at the level of Bickel and Doksum (1977), some understanding of the Bayesian approach as in Box and Tiao (1973), experience with condi tional inference at the level of Cox and Snell (1989) and exposure to statistical models as found in McCullagh and Neider (1989). I have chosen not to present the proofs of convergence or rates of convergence since these proofs may require substantial background in Markov chain theory which is beyond the scope ofthis book. However, references to these proofs are given. There has been an explosion of papers in the area of Markov chain Monte Carlo in the last five years. I have attempted to identify key references - though due to the volatility of the field some work may have been missed.
Author: R. J. Serfling
Publisher:
ISBN: 9780471137306
Category : Limit theorems (Probability theory)
Languages : en
Pages : 371
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Book Description
Author: Louis H.Y. Chen
Publisher: Springer Science & Business Media
ISBN: 3642150071
Category : Mathematics
Languages : en
Pages : 411
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Book Description
Since its introduction in 1972, Stein’s method has offered a completely novel way of evaluating the quality of normal approximations. Through its characterizing equation approach, it is able to provide approximation error bounds in a wide variety of situations, even in the presence of complicated dependence. Use of the method thus opens the door to the analysis of random phenomena arising in areas including statistics, physics, and molecular biology. Though Stein's method for normal approximation is now mature, the literature has so far lacked a complete self contained treatment. This volume contains thorough coverage of the method’s fundamentals, includes a large number of recent developments in both theory and applications, and will help accelerate the appreciation, understanding, and use of Stein's method by providing the reader with the tools needed to apply it in new situations. It addresses researchers as well as graduate students in Probability, Statistics and Combinatorics.
Author: Peter McCullagh
Publisher: Courier Dover Publications
ISBN: 0486832694
Category : Mathematics
Languages : en
Pages : 308
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Book Description
A pioneering monograph on tensor methods applied to distributional problems arising in statistics, this work begins with the study of multivariate moments and cumulants. An invaluable reference for graduate students and professional statisticians. 1987 edition.
Author: John Edward Kolassa
Publisher: Springer Science & Business Media
ISBN: 9780387942773
Category : Mathematics
Languages : en
Pages : 150
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Book Description
Asymptotic techniques have long been important in statistical inference; these techniques remain important in the age of fast computing because some exact answers are still either conceptually unavailable or practically out of reach. This book presents theoretical results relevant to Edgeworth and saddlepoint expansions to densities and distribution functions. It provides examples of their application in some simple, and in a few complicated, settings. Numerical and asymptotic assessments of accuracy are presented. Variants of these expansions, including much of modern likelihood theory, are discussed. Applications to lattice distributions are extensively treated.
Author: Harold Cohen
Publisher: Springer Science & Business Media
ISBN: 1441998365
Category : Mathematics
Languages : en
Pages : 493
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Book Description
This book presents numerical and other approximation techniques for solving various types of mathematical problems that cannot be solved analytically. In addition to well known methods, it contains some non-standard approximation techniques that are now formally collected as well as original methods developed by the author that do not appear in the literature. This book contains an extensive treatment of approximate solutions to various types of integral equations, a topic that is not often discussed in detail. There are detailed analyses of ordinary and partial differential equations and descriptions of methods for estimating the values of integrals that are presented in a level of detail that will suggest techniques that will be useful for developing methods for approximating solutions to problems outside of this text. The book is intended for researchers who must approximate solutions to problems that cannot be solved analytically. It is also appropriate for students taking courses in numerical approximation techniques.
Author: Ronald W. Butler
Publisher: Cambridge University Press
ISBN: 1139466518
Category : Mathematics
Languages : en
Pages : 548
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Book Description
Modern statistical methods use complex, sophisticated models that can lead to intractable computations. Saddlepoint approximations can be the answer. Written from the user's point of view, this book explains in clear language how such approximate probability computations are made, taking readers from the very beginnings to current applications. The core material is presented in chapters 1-6 at an elementary mathematical level. Chapters 7-9 then give a highly readable account of higher-order asymptotic inference. Later chapters address areas where saddlepoint methods have had substantial impact: multivariate testing, stochastic systems and applied probability, bootstrap implementation in the transform domain, and Bayesian computation and inference. No previous background in the area is required. Data examples from real applications demonstrate the practical value of the methods. Ideal for graduate students and researchers in statistics, biostatistics, electrical engineering, econometrics, and applied mathematics, this is both an entry-level text and a valuable reference.
Author: A. W. van der Vaart
Publisher: Cambridge University Press
ISBN: 9780521784504
Category : Mathematics
Languages : en
Pages : 470
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Book Description
This book is an introduction to the field of asymptotic statistics. The treatment is both practical and mathematically rigorous. In addition to most of the standard topics of an asymptotics course, including likelihood inference, M-estimation, the theory of asymptotic efficiency, U-statistics, and rank procedures, the book also presents recent research topics such as semiparametric models, the bootstrap, and empirical processes and their applications. The topics are organized from the central idea of approximation by limit experiments, which gives the book one of its unifying themes. This entails mainly the local approximation of the classical i.i.d. set up with smooth parameters by location experiments involving a single, normally distributed observation. Thus, even the standard subjects of asymptotic statistics are presented in a novel way. Suitable as a graduate or Master s level statistics text, this book will also give researchers an overview of the latest research in asymptotic statistics.
