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Author: Shailaja Deshmukh
Publisher: Springer Nature
ISBN: 9811590036
Category : Mathematics
Languages : en
Pages : 540
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Book Description
The book presents the fundamental concepts from asymptotic statistical inference theory, elaborating on some basic large sample optimality properties of estimators and some test procedures. The most desirable property of consistency of an estimator and its large sample distribution, with suitable normalization, are discussed, the focus being on the consistent and asymptotically normal (CAN) estimators. It is shown that for the probability models belonging to an exponential family and a Cramer family, the maximum likelihood estimators of the indexing parameters are CAN. The book describes some large sample test procedures, in particular, the most frequently used likelihood ratio test procedure. Various applications of the likelihood ratio test procedure are addressed, when the underlying probability model is a multinomial distribution. These include tests for the goodness of fit and tests for contingency tables. The book also discusses a score test and Wald’s test, their relationship with the likelihood ratio test and Karl Pearson’s chi-square test. An important finding is that, while testing any hypothesis about the parameters of a multinomial distribution, a score test statistic and Karl Pearson’s chi-square test statistic are identical. Numerous illustrative examples of differing difficulty level are incorporated to clarify the concepts. For better assimilation of the notions, various exercises are included in each chapter. Solutions to almost all the exercises are given in the last chapter, to motivate students towards solving these exercises and to enable digestion of the underlying concepts. The concepts from asymptotic inference are crucial in modern statistics, but are difficult to grasp in view of their abstract nature. To overcome this difficulty, keeping up with the recent trend of using R software for statistical computations, the book uses it extensively, for illustrating the concepts, verifying the properties of estimators and carrying out various test procedures. The last section of the chapters presents R codes to reveal and visually demonstrate the hidden aspects of different concepts and procedures. Augmenting the theory with R software is a novel and a unique feature of the book. The book is designed primarily to serve as a text book for a one semester introductory course in asymptotic statistical inference, in a post-graduate program, such as Statistics, Bio-statistics or Econometrics. It will also provide sufficient background information for studying inference in stochastic processes. The book will cater to the need of a concise but clear and student-friendly book introducing, conceptually and computationally, basics of asymptotic inference.
Author: Shailaja Deshmukh
Publisher: Springer Nature
ISBN: 9811590036
Category : Mathematics
Languages : en
Pages : 540
Get Book Here
Book Description
The book presents the fundamental concepts from asymptotic statistical inference theory, elaborating on some basic large sample optimality properties of estimators and some test procedures. The most desirable property of consistency of an estimator and its large sample distribution, with suitable normalization, are discussed, the focus being on the consistent and asymptotically normal (CAN) estimators. It is shown that for the probability models belonging to an exponential family and a Cramer family, the maximum likelihood estimators of the indexing parameters are CAN. The book describes some large sample test procedures, in particular, the most frequently used likelihood ratio test procedure. Various applications of the likelihood ratio test procedure are addressed, when the underlying probability model is a multinomial distribution. These include tests for the goodness of fit and tests for contingency tables. The book also discusses a score test and Wald’s test, their relationship with the likelihood ratio test and Karl Pearson’s chi-square test. An important finding is that, while testing any hypothesis about the parameters of a multinomial distribution, a score test statistic and Karl Pearson’s chi-square test statistic are identical. Numerous illustrative examples of differing difficulty level are incorporated to clarify the concepts. For better assimilation of the notions, various exercises are included in each chapter. Solutions to almost all the exercises are given in the last chapter, to motivate students towards solving these exercises and to enable digestion of the underlying concepts. The concepts from asymptotic inference are crucial in modern statistics, but are difficult to grasp in view of their abstract nature. To overcome this difficulty, keeping up with the recent trend of using R software for statistical computations, the book uses it extensively, for illustrating the concepts, verifying the properties of estimators and carrying out various test procedures. The last section of the chapters presents R codes to reveal and visually demonstrate the hidden aspects of different concepts and procedures. Augmenting the theory with R software is a novel and a unique feature of the book. The book is designed primarily to serve as a text book for a one semester introductory course in asymptotic statistical inference, in a post-graduate program, such as Statistics, Bio-statistics or Econometrics. It will also provide sufficient background information for studying inference in stochastic processes. The book will cater to the need of a concise but clear and student-friendly book introducing, conceptually and computationally, basics of asymptotic inference.
Author: Masanobu Taniguchi
Publisher: Springer
ISBN: 9781461270287
Category : Mathematics
Languages : en
Pages : 0
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Book Description
The primary aim of this book is to provide modern statistical techniques and theory for stochastic processes. The stochastic processes mentioned here are not restricted to the usual AR, MA, and ARMA processes. A wide variety of stochastic processes, including non-Gaussian linear processes, long-memory processes, nonlinear processes, non-ergodic processes and diffusion processes are described. The authors discuss estimation and testing theory and many other relevant statistical methods and techniques.
Author: Masahito Hayashi
Publisher: World Scientific
ISBN: 981448198X
Category : Science
Languages : en
Pages : 553
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Book Description
Quantum statistical inference, a research field with deep roots in the foundations of both quantum physics and mathematical statistics, has made remarkable progress since 1990. In particular, its asymptotic theory has been developed during this period. However, there has hitherto been no book covering this remarkable progress after 1990; the famous textbooks by Holevo and Helstrom deal only with research results in the earlier stage (1960s-1970s).This book presents the important and recent results of quantum statistical inference. It focuses on the asymptotic theory, which is one of the central issues of mathematical statistics and had not been investigated in quantum statistical inference until the early 1980s. It contains outstanding papers after Holevo's textbook, some of which are of great importance but are not available now.The reader is expected to have only elementary mathematical knowledge, and therefore much of the content will be accessible to graduate students as well as research workers in related fields. Introductions to quantum statistical inference have been specially written for the book. Asymptotic Theory of Quantum Statistical Inference: Selected Papers will give the reader a new insight into physics and statistical inference.
Author: B. L. S. Prakasa Rao
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 458
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Book Description
Probability and stochastic processes; Limit theorems for some statistics; Asymptotic theory of estimation; Linear parametric inference; Martingale approach to inference; Inference in nonlinear regression; Von mises functionals; Empirical characteristic function and its applications.
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.
Author: Lucien Le Cam
Publisher: Springer Science & Business Media
ISBN: 1461211662
Category : Mathematics
Languages : en
Pages : 299
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Book Description
This is the second edition of a coherent introduction to the subject of asymptotic statistics as it has developed over the past 50 years. It differs from the first edition in that it is now more 'reader friendly' and also includes a new chapter on Gaussian and Poisson experiments, reflecting their growing role in the field. Most of the subsequent chapters have been entirely rewritten and the nonparametrics of Chapter 7 have been amplified. The volume is not intended to replace monographs on specialized subjects, but will help to place them in a coherent perspective. It thus represents a link between traditional material - such as maximum likelihood, and Wald's Theory of Statistical Decision Functions -- together with comparison and distances for experiments. Much of the material has been taught in a second year graduate course at Berkeley for 30 years.
Author: Anirban DasGupta
Publisher: Springer Science & Business Media
ISBN: 0387759700
Category : Mathematics
Languages : en
Pages : 726
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Book Description
This unique book delivers an encyclopedic treatment of classic as well as contemporary large sample theory, dealing with both statistical problems and probabilistic issues and tools. The book is unique in its detailed coverage of fundamental topics. 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. There is no other book in large sample theory that matches this book in coverage, exercises and examples, bibliography, and lucid conceptual discussion of issues and theorems.
Author: Pranab K. Sen
Publisher: Cambridge University Press
ISBN: 0521877229
Category : Mathematics
Languages : en
Pages : 399
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Book Description
A broad view of exact statistical inference and the development of asymptotic statistical inference.
Author: J. Pfanzagl
Publisher: Springer Science & Business Media
ISBN: 1461257697
Category : Mathematics
Languages : en
Pages : 324
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Book Description
Author: I.A. Ibragimov
Publisher: Springer Science & Business Media
ISBN: 1489900276
Category : Mathematics
Languages : en
Pages : 410
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Book Description
when certain parameters in the problem tend to limiting values (for example, when the sample size increases indefinitely, the intensity of the noise ap proaches zero, etc.) To address the problem of asymptotically optimal estimators consider the following important case. Let X 1, X 2, ... , X n be independent observations with the joint probability density !(x,O) (with respect to the Lebesgue measure on the real line) which depends on the unknown patameter o e 9 c R1. It is required to derive the best (asymptotically) estimator 0:( X b ... , X n) of the parameter O. The first question which arises in connection with this problem is how to compare different estimators or, equivalently, how to assess their quality, in terms of the mean square deviation from the parameter or perhaps in some other way. The presently accepted approach to this problem, resulting from A. Wald's contributions, is as follows: introduce a nonnegative function w(0l> ( ), Ob Oe 9 (the loss function) and given two estimators Of and O! n 2 2 the estimator for which the expected loss (risk) Eown(Oj, 0), j = 1 or 2, is smallest is called the better with respect to Wn at point 0 (here EoO is the expectation evaluated under the assumption that the true value of the parameter is 0). Obviously, such a method of comparison is not without its defects.
