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| weak convergence and empirical processes: Weak Convergence and Empirical Processes AW van der Vaart, Aad van der Vaart, Jon Wellner, 1996-03-14 This book explores weak convergence theory and empirical processes and their applications to many applications in statistics. Part one reviews stochastic convergence in its various forms. Part two offers the theory of empirical processes in a form accessible to statisticians and probabilists. Part three covers a range of topics demonstrating the applicability of the theory to key questions such as measures of goodness of fit and the bootstrap. |
| weak convergence and empirical processes: Weak Convergence and Empirical Processes A. W. van der Vaart, Jon A. Wellner, 2023 This book provides an account of weak convergence theory, empirical processes, and their application to a wide variety of problems in statistics. The first part of the book presents a thorough treatment of stochastic convergence in its various forms. Part 2 brings together the theory of empirical processes in a form accessible to statisticians and probabilists. In Part 3, the authors cover a range of applications in statistics including rates of convergence of estimators; limit theorems for M− and Z−estimators; the bootstrap; the functional delta-method and semiparametric estimation. Most of the chapters conclude with problems and complements. Some of these are exercises to help the reader's understanding of the material, whereas others are intended to supplement the text. This second edition includes many of the new developments in the field since publication of the first edition in 1996: Glivenko-Cantelli preservation theorems; new bounds on expectations of suprema of empirical processes; new bounds on covering numbers for various function classes; generic chaining; definitive versions of concentration bounds; and new applications in statistics including penalized M-estimation, the lasso, classification, and support vector machines. The approximately 200 additional pages also round out classical subjects, including chapters on weak convergence in Skorokhod space, on stable convergence, and on processes based on pseudo-observations. |
| weak convergence and empirical processes: Empirical Processes with Applications to Statistics Galen R. Shorack, Jon A. Wellner, 2009-01-01 Originally published in 1986, this valuable reference provides a detailed treatment of limit theorems and inequalities for empirical processes of real-valued random variables; applications of the theory to censored data, spacings, rank statistics, quantiles, and many functionals of empirical processes, including a treatment of bootstrap methods; and a summary of inequalities that are useful for proving limit theorems. At the end of the Errata section, the authors have supplied references to solutions for 11 of the 19 Open Questions provided in the book's original edition. Audience: researchers in statistical theory, probability theory, biostatistics, econometrics, and computer science. |
| weak convergence and empirical processes: On Weak Convergence of Empirical Processes for Random Number of Independent Stochastic Vectors Pranab Kumar Sen, 1971 |
| weak convergence and empirical processes: Introduction to Empirical Processes and Semiparametric Inference Michael R. Kosorok, 2007-12-29 Kosorok’s brilliant text provides a self-contained introduction to empirical processes and semiparametric inference. These powerful research techniques are surprisingly useful for developing methods of statistical inference for complex models and in understanding the properties of such methods. This is an authoritative text that covers all the bases, and also a friendly and gradual introduction to the area. The book can be used as research reference and textbook. |
| weak convergence and empirical processes: Convergence of Stochastic Processes D. Pollard, 1984-10-08 Functionals on stochastic processes; Uniform convergence of empirical measures; Convergence in distribution in euclidean spaces; Convergence in distribution in metric spaces; The uniform metric on space of cadlag functions; The skorohod metric on D [0, oo); Central limit teorems; Martingales. |
| weak convergence and empirical processes: Weak Convergence and Empirical Processes Aad van der vaart, Jon Wellner, 2013-03-09 This book explores weak convergence theory and empirical processes and their applications to many applications in statistics. Part one reviews stochastic convergence in its various forms. Part two offers the theory of empirical processes in a form accessible to statisticians and probabilists. Part three covers a range of topics demonstrating the applicability of the theory to key questions such as measures of goodness of fit and the bootstrap. |
| weak convergence and empirical processes: Weighted Empirical Processes in Dynamic Nonlinear Models Hira L. Koul, 2002-06-13 This book presents a unified approach for obtaining the limiting distributions of minimum distance. It discusses classes of goodness-of-t tests for fitting an error distribution in some of these models and/or fitting a regression-autoregressive function without assuming the knowledge of the error distribution. The main tool is the asymptotic equi-continuity of certain basic weighted residual empirical processes in the uniform and L2 metrics. |
| weak convergence and empirical processes: Weak Convergence and Its Applications Zhengyan Lin, Hanchao Wang, 2014 Weak convergence of stochastic processes is one of most important theories in probability theory. Not only probability experts but also more and more statisticians are interested in it. In the study of statistics and econometrics, some problems cannot be solved by the classical method. In this book, we will introduce some recent development of modern weak convergence theory to overcome defects of classical theory.Contents: The Definition and Basic Properties of Weak Convergence: Metric SpaceThe Definition of Weak Convergence of Stochastic Processes and Portmanteau TheoremHow to Verify the Weak Convergence?Two Examples of Applications of Weak ConvergenceConvergence to the Independent Increment Processes: The Basic Conditions of Convergence to the Gaussian Independent Increment ProcessesDonsker Invariance PrincipleConvergence of Poisson Point ProcessesTwo Examples of Applications of Point Process MethodConvergence to Semimartingales: The Conditions of Tightness for Semimartingale SequenceWeak Convergence to SemimartingaleWeak Convergence to Stochastic Integral I: The Martingale Convergence ApproachWeak Convergence to Stochastic Integral II: Kurtz and Protter's ApproachStable Central Limit Theorem for SemimartingalesAn Application to Stochastic Differential EquationsAppendix: The Predictable Characteristics of SemimartingalesConvergence of Empirical Processes: Classical Weak Convergence of Empirical ProcessesWeak Convergence of Marked Empirical ProcessesWeak Convergence of Function Index Empirical ProcessesWeak Convergence of Empirical Processes Involving Time-Dependent dataTwo Examples of Applications in Statistics Readership: Graduate students and researchers in probability & statistics and econometrics. |
| weak convergence and empirical processes: A Weak Convergence Approach to the Theory of Large Deviations Paul Dupuis, Richard S. Ellis, 2011-09-09 Applies the well-developed tools of the theory of weak convergenceof probability measures to large deviation analysis--a consistentnew approach The theory of large deviations, one of the most dynamic topics inprobability today, studies rare events in stochastic systems. Thenonlinear nature of the theory contributes both to its richness anddifficulty. This innovative text demonstrates how to employ thewell-established linear techniques of weak convergence theory toprove large deviation results. Beginning with a step-by-stepdevelopment of the approach, the book skillfully guides readersthrough models of increasing complexity covering a wide variety ofrandom variable-level and process-level problems. Representationformulas for large deviation-type expectations are a key tool andare developed systematically for discrete-time problems. Accessible to anyone who has a knowledge of measure theory andmeasure-theoretic probability, A Weak Convergence Approach to theTheory of Large Deviations is important reading for both studentsand researchers. |
| weak convergence and empirical processes: Principles of Nonparametric Learning Laszlo Györfi, 2014-05-04 This volume provides a systematic in-depth analysis of nonparametric learning. It covers the theoretical limits and the asymptotical optimal algorithms and estimates, such as pattern recognition, nonparametric regression estimation, universal prediction, vector quantization, distribution and density estimation, and genetic programming. |
| weak convergence and empirical processes: Weak Convergence and Empirical Processes A. W. van der Vaart, Jon A. Wellner, 2023-07-11 This book provides an account of weak convergence theory, empirical processes, and their application to a wide variety of problems in statistics. The first part of the book presents a thorough treatment of stochastic convergence in its various forms. Part 2 brings together the theory of empirical processes in a form accessible to statisticians and probabilists. In Part 3, the authors cover a range of applications in statistics including rates of convergence of estimators; limit theorems for M− and Z−estimators; the bootstrap; the functional delta-method and semiparametric estimation. Most of the chapters conclude with “problems and complements.” Some of these are exercises to help the reader’s understanding of the material, whereas others are intended to supplement the text. This second edition includes many of the new developments in the field since publication of the first edition in 1996: Glivenko-Cantelli preservation theorems; new bounds on expectations of suprema of empirical processes; new bounds on covering numbers for various function classes; generic chaining; definitive versions of concentration bounds; and new applications in statistics including penalized M-estimation, the lasso, classification, and support vector machines. The approximately 200 additional pages also round out classical subjects, including chapters on weak convergence in Skorokhod space, on stable convergence, and on processes based on pseudo-observations. |
| weak convergence and empirical processes: Empirical Processes David Pollard, 1990 |
| weak convergence and empirical processes: Weak Convergence of Stochastic Processes Vidyadhar Mandrekar, 2016 The purpose of this book is to present results on the subject of weak convergence to study invariance principles in statistical applications. Different techniques, formerly only available in a broad range of literature, are for the first time presen |
| weak convergence and empirical processes: Analysis and Approximation of Rare Events Amarjit Budhiraja, Paul Dupuis, 2019-08-10 This book presents broadly applicable methods for the large deviation and moderate deviation analysis of discrete and continuous time stochastic systems. A feature of the book is the systematic use of variational representations for quantities of interest such as normalized logarithms of probabilities and expected values. By characterizing a large deviation principle in terms of Laplace asymptotics, one converts the proof of large deviation limits into the convergence of variational representations. These features are illustrated though their application to a broad range of discrete and continuous time models, including stochastic partial differential equations, processes with discontinuous statistics, occupancy models, and many others. The tools used in the large deviation analysis also turn out to be useful in understanding Monte Carlo schemes for the numerical approximation of the same probabilities and expected values. This connection is illustrated through the design and analysis of importance sampling and splitting schemes for rare event estimation. The book assumes a solid background in weak convergence of probability measures and stochastic analysis, and is suitable for advanced graduate students, postdocs and researchers. |
| weak convergence and empirical processes: High Dimensional Probability II Evarist Giné, David M. Mason, Jon A. Wellner, 2012-12-06 High dimensional probability, in the sense that encompasses the topics rep resented in this volume, began about thirty years ago with research in two related areas: limit theorems for sums of independent Banach space valued random vectors and general Gaussian processes. An important feature in these past research studies has been the fact that they highlighted the es sential probabilistic nature of the problems considered. In part, this was because, by working on a general Banach space, one had to discard the extra, and often extraneous, structure imposed by random variables taking values in a Euclidean space, or by processes being indexed by sets in R or Rd. Doing this led to striking advances, particularly in Gaussian process theory. It also led to the creation or introduction of powerful new tools, such as randomization, decoupling, moment and exponential inequalities, chaining, isoperimetry and concentration of measure, which apply to areas well beyond those for which they were created. The general theory of em pirical processes, with its vast applications in statistics, the study of local times of Markov processes, certain problems in harmonic analysis, and the general theory of stochastic processes are just several of the broad areas in which Gaussian process techniques and techniques from probability in Banach spaces have made a substantial impact. Parallel to this work on probability in Banach spaces, classical proba bility and empirical process theory were enriched by the development of powerful results in strong approximations. |
| weak convergence and empirical processes: A Weak Convergence Approach to the Theory of Large Deviations Paul Dupuis, Richard S. Ellis, 1997-02-27 Applies the well-developed tools of the theory of weak convergenceof probability measures to large deviation analysis--a consistentnew approach The theory of large deviations, one of the most dynamic topics inprobability today, studies rare events in stochastic systems. Thenonlinear nature of the theory contributes both to its richness anddifficulty. This innovative text demonstrates how to employ thewell-established linear techniques of weak convergence theory toprove large deviation results. Beginning with a step-by-stepdevelopment of the approach, the book skillfully guides readersthrough models of increasing complexity covering a wide variety ofrandom variable-level and process-level problems. Representationformulas for large deviation-type expectations are a key tool andare developed systematically for discrete-time problems. Accessible to anyone who has a knowledge of measure theory andmeasure-theoretic probability, A Weak Convergence Approach to theTheory of Large Deviations is important reading for both studentsand researchers. |
| weak convergence and empirical processes: Weak Convergence of Weighted Empirical Processes Under Long Range Dependence with Applications to Robust Estimation in Linear Models Kanchan Mukherjee, 1993 |
| weak convergence and empirical processes: Long-Memory Processes Jan Beran, Yuanhua Feng, Sucharita Ghosh, Rafal Kulik, 2013-05-14 Long-memory processes are known to play an important part in many areas of science and technology, including physics, geophysics, hydrology, telecommunications, economics, finance, climatology, and network engineering. In the last 20 years enormous progress has been made in understanding the probabilistic foundations and statistical principles of such processes. This book provides a timely and comprehensive review, including a thorough discussion of mathematical and probabilistic foundations and statistical methods, emphasizing their practical motivation and mathematical justification. Proofs of the main theorems are provided and data examples illustrate practical aspects. This book will be a valuable resource for researchers and graduate students in statistics, mathematics, econometrics and other quantitative areas, as well as for practitioners and applied researchers who need to analyze data in which long memory, power laws, self-similar scaling or fractal properties are relevant. |
| weak convergence and empirical processes: Probability in Banach Spaces Michel Ledoux, Michel Talagrand, 2013-03-09 Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed. |
| weak convergence and empirical processes: High-Dimensional Probability Roman Vershynin, 2018-09-27 An integrated package of powerful probabilistic tools and key applications in modern mathematical data science. |
| weak convergence and empirical processes: Quantile Processes with Statistical Applications Miklos Csorgo, 1983-01-31 Provides a comprehensive theory of the approximations of quantile processes as well as some of their statistical applications. |
| weak convergence and empirical processes: Unified Methods for Censored Longitudinal Data and Causality Mark J. van der Laan, James M Robins, 2012-11-12 A fundamental statistical framework for the analysis of complex longitudinal data is provided in this book. It provides the first comprehensive description of optimal estimation techniques based on time-dependent data structures. The techniques go beyond standard statistical approaches and can be used to teach masters and Ph.D. students. The text is ideally suitable for researchers in statistics with a strong interest in the analysis of complex longitudinal data. |
| weak convergence and empirical processes: Empirical Processes in M-Estimation Sara A. Geer, 2000-01-28 Advanced text; estimation methods in statistics, e.g. least squares; lots of examples; minimal abstraction. |
| weak convergence and empirical processes: Uniform Central Limit Theorems R. M. Dudley, 1999-07-28 This treatise by an acknowledged expert includes several topics not found in any previous book. |
| weak convergence and empirical processes: Upper and Lower Bounds for Stochastic Processes Michel Talagrand, 2022-01-01 This book provides an in-depth account of modern methods used to bound the supremum of stochastic processes. Starting from first principles, it takes the reader to the frontier of current research. This second edition has been completely rewritten, offering substantial improvements to the exposition and simplified proofs, as well as new results. The book starts with a thorough account of the generic chaining, a remarkably simple and powerful method to bound a stochastic process that should belong to every probabilist’s toolkit. The effectiveness of the scheme is demonstrated by the characterization of sample boundedness of Gaussian processes. Much of the book is devoted to exploring the wealth of ideas and results generated by thirty years of efforts to extend this result to more general classes of processes, culminating in the recent solution of several key conjectures. A large part of this unique book is devoted to the author’s influential work. While many of the results presented are rather advanced, others bear on the very foundations of probability theory. In addition to providing an invaluable reference for researchers, the book should therefore also be of interest to a wide range of readers. |
| weak convergence and empirical processes: Asymptotic Statistics A. W. van der Vaart, 2000-06-19 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. |
| weak convergence and empirical processes: Weak Convergence (IA) Tchilabalo Abozou Kpanzou, Modou Ngom, Aladji Babacar Niang, 2021-11-08 This monograph aims at presenting the core weak convergence theory for sequences of random vectors with values in dimension k. In some places, a more general formulation in metric spaces is provided. It lays out the necessary foundation that paves the way to applications in particular sub-fields of the theory. In particular, the needs of Asymptotic Statistics are addressed. A whole chapter is devoted to weak convergence in the real line where specific tools, for example for handling weak convergence of sequences using independent and identically distributed random variables such that the Renyi's representations by means of standard uniform or exponential random variables, are stated. The functional empirical process is presented as a powerful tool for solving a considerable number of asymptotic problems in Statistics. The text is written in a self-contained approach with the proofs of all used results at the exception of the general Skorohod-Wichura Theorem. We finish the book with a chapter on weak convergence of bounded measures and locally bounded measures in preparation of a more general theory of measures on topological spaces |
| weak convergence and empirical processes: Approximate Distributions of Order Statistics Rolf-Dieter Reiss, 2012-12-06 This book is designed as a unified and mathematically rigorous treatment of some recent developments of the asymptotic distribution theory of order statistics (including the extreme order statistics) that are relevant for statistical theory and its applications. Particular emphasis is placed on results concern ing the accuracy oflimit theorems, on higher order approximations, and other approximations in quite a general sense. Contrary to the classical limit theorems that primarily concern the weak convergence of distribution functions, our main results will be formulated in terms of the variational and the Hellinger distance. These results will form the proper springboard for the investigation of parametric approximations of nonparametric models of joint distributions of order statistics. The approxi mating models include normal as well as extreme value models. Several applications will show the usefulness of this approach. Other recent developments in statistics like nonparametric curve estima tion and the bootstrap method will be studied as far as order statistics are concerned. 1n connection with this, graphical methods will, to some extent, be explored. |
| weak convergence and empirical processes: Modern Multidimensional Scaling Ingwer Borg, Patrick Groenen, 2013-04-18 Multidimensional scaling (MDS) is a technique for the analysis of similarity or dissimilarity data on a set of objects. Such data may be intercorrelations of test items, ratings of similarity on political candidates, or trade indices for a set of countries. MDS attempts to model such data as distances among points in a geometric space. The main reason for doing this is that one wants a graphical display of the structure of the data, one that is much easier to understand than an array of numbers and, moreover, one that displays the essential information in the data, smoothing out noise. There are numerous varieties of MDS. Some facets for distinguishing among them are the particular type of geometry into which one wants to map the data, the mapping function, the algorithms used to find an optimal data representation, the treatment of statistical error in the models, or the possibility to represent not just one but several similarity matrices at the same time. Other facets relate to the different purposes for which MDS has been used, to various ways of looking at or interpreting an MDS representation, or to differences in the data required for the particular models. In this book, we give a fairly comprehensive presentation of MDS. For the reader with applied interests only, the first six chapters of Part I should be sufficient. They explain the basic notions of ordinary MDS, with an emphasis on how MDS can be helpful in answering substantive questions. |
| weak convergence and empirical processes: The Bootstrap and Edgeworth Expansion Peter Hall, 2013-12-01 This monograph addresses two quite different topics, each being able to shed light on the other. Firstly, it lays the foundation for a particular view of the bootstrap. Secondly, it gives an account of Edgeworth expansion. The first two chapters deal with the bootstrap and Edgeworth expansion respectively, while chapters 3 and 4 bring these two themes together, using Edgeworth expansion to explore and develop the properties of the bootstrap. The book is aimed at graduate level for those with some exposure to the methods of theoretical statistics. However, technical details are delayed until the last chapter such that mathematically able readers without knowledge of the rigorous theory of probability will have no trouble understanding most of the book. |
| weak convergence and empirical processes: Test Equating Michael J. Kolen, Robert L. Brennan, 2013-11-11 In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques. This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved. The main themes are: - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating. The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented. As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for the first time as part of their graduate study will find this an invaluable text and reference. |
| weak convergence and empirical processes: Asymptotic Methods in Statistical Decision Theory Lucien Le Cam, 2012-12-06 This book grew out of lectures delivered at the University of California, Berkeley, over many years. The subject is a part of asymptotics in statistics, organized around a few central ideas. The presentation proceeds from the general to the particular since this seemed the best way to emphasize the basic concepts. The reader is expected to have been exposed to statistical thinking and methodology, as expounded for instance in the book by H. Cramer [1946] or the more recent text by P. Bickel and K. Doksum [1977]. Another pos sibility, closer to the present in spirit, is Ferguson [1967]. Otherwise the reader is expected to possess some mathematical maturity, but not really a great deal of detailed mathematical knowledge. Very few mathematical objects are used; their assumed properties are simple; the results are almost always immediate consequences of the definitions. Some objects, such as vector lattices, may not have been included in the standard background of a student of statistics. For these we have provided a summary of relevant facts in the Appendix. The basic structures in the whole affair are systems that Blackwell called experiments and transitions between them. An experiment is a mathe matical abstraction intended to describe the basic features of an observational process if that process is contemplated in advance of its implementation. Typically, an experiment consists of a set E> of theories about what may happen in the observational process. |
| weak convergence and empirical processes: Statistical Methods for Spatio-Temporal Systems Barbel Finkenstadt, Leonhard Held, Valerie Isham, 2006-10-20 Statistical Methods for Spatio-Temporal Systems presents current statistical research issues on spatio-temporal data modeling and will promote advances in research and a greater understanding between the mechanistic and the statistical modeling communities. Contributed by leading researchers in the field, each self-contained chapter starts w |
| weak convergence and empirical processes: Supersymmetry and Equivariant de Rham Theory Victor W Guillemin, Shlomo Sternberg, 2013-03-09 This book discusses the equivariant cohomology theory of differentiable manifolds. Although this subject has gained great popularity since the early 1980's, it has not before been the subject of a monograph. It covers almost all important aspects of the subject The authors are key authorities in this field. |
| weak convergence and empirical processes: Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems Vladimir Koltchinskii, 2011-07-29 The purpose of these lecture notes is to provide an introduction to the general theory of empirical risk minimization with an emphasis on excess risk bounds and oracle inequalities in penalized problems. In recent years, there have been new developments in this area motivated by the study of new classes of methods in machine learning such as large margin classification methods (boosting, kernel machines). The main probabilistic tools involved in the analysis of these problems are concentration and deviation inequalities by Talagrand along with other methods of empirical processes theory (symmetrization inequalities, contraction inequality for Rademacher sums, entropy and generic chaining bounds). Sparse recovery based on l_1-type penalization and low rank matrix recovery based on the nuclear norm penalization are other active areas of research, where the main problems can be stated in the framework of penalized empirical risk minimization, and concentration inequalities and empirical processes tools have proved to be very useful. |
| weak convergence and empirical processes: Approximation of Functions G. G. Lorentz, 2023-05-08 This is an easily accessible account of the approximation of functions. It is simple and without unnecessary details, but complete enough to include the classical results of the theory. With only a few exceptions, only functions of one real variable are considered. A major theme is the degree of uniform approximation by linear sets of functions. This encompasses approximations by trigonometric polynomials, algebraic polynomials, rational functions, and polynomial operators. The chapter on approximation by operators does not assume extensive knowledge of functional analysis. Two chapters cover the important topics of widths and entropy. The last chapter covers the solution by Kolmogorov and Arnol?d of Hilbert's 13th problem. There are notes at the end of each chapter that give information about important topics not treated in the main text. Each chapter also has a short set of challenging problems, which serve as illustrations. |
| weak convergence and empirical processes: Concentration Inequalities Stéphane Boucheron, Gábor Lugosi, Pascal Massart, 2013-02-07 Describes the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities to transportation arguments to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are also presented. |
| weak convergence and empirical processes: Applied Probability Kenneth Lange, 2008-01-17 Despite the fears of university mathematics departments, mathematics educat,ion is growing rather than declining. But the truth of the matter is that the increases are occurring outside departments of mathematics. Engineers, computer scientists, physicists, chemists, economists, statis- cians, biologists, and even philosophers teach and learn a great deal of mathematics. The teaching is not always terribly rigorous, but it tends to be better motivated and better adapted to the needs of students. In my own experience teaching students of biostatistics and mathematical bi- ogy, I attempt to convey both the beauty and utility of probability. This is a tall order, partially because probability theory has its own vocabulary and habits of thought. The axiomatic presentation of advanced probability typically proceeds via measure theory. This approach has the advantage of rigor, but it inwitably misses most of the interesting applications, and many applied scientists rebel against the onslaught of technicalities. In the current book, I endeavor to achieve a balance between theory and app- cations in a rather short compass. While the combination of brevity apd balance sacrifices many of the proofs of a rigorous course, it is still cons- tent with supplying students with many of the relevant theoretical tools. In my opinion, it better to present the mathematical facts without proof rather than omit them altogether. |
| weak convergence and empirical processes: Empirical Process Techniques for Dependent Data Herold Dehling, Thomas Mikosch, Michael Sörensen, 2012-12-06 Empirical process techniques for independent data have been used for many years in statistics and probability theory. These techniques have proved very useful for studying asymptotic properties of parametric as well as non-parametric statistical procedures. Recently, the need to model the dependence structure in data sets from many different subject areas such as finance, insurance, and telecommunications has led to new developments concerning the empirical distribution function and the empirical process for dependent, mostly stationary sequences. This work gives an introduction to this new theory of empirical process techniques, which has so far been scattered in the statistical and probabilistic literature, and surveys the most recent developments in various related fields. Key features: A thorough and comprehensive introduction to the existing theory of empirical process techniques for dependent data * Accessible surveys by leading experts of the most recent developments in various related fields * Examines empirical process techniques for dependent data, useful for studying parametric and non-parametric statistical procedures * Comprehensive bibliographies * An overview of applications in various fields related to empirical processes: e.g., spectral analysis of time-series, the bootstrap for stationary sequences, extreme value theory, and the empirical process for mixing dependent observations, including the case of strong dependence. To date this book is the only comprehensive treatment of the topic in book literature. It is an ideal introductory text that will serve as a reference or resource for classroom use in the areas of statistics, time-series analysis, extreme value theory, point process theory, and applied probability theory. Contributors: P. Ango Nze, M.A. Arcones, I. Berkes, R. Dahlhaus, J. Dedecker, H.G. Dehling, |
Weak Convergence and Empirical Processes - Springer
About this book. This book provides an account of weak convergence theory, empirical processes, and their application to a wide variety of problems in statistics. The first part of the book presents a thorough treatment of stochastic convergence in its various forms. Part 2 brings together the theory of empirical processes in a form accessible ...
[PDF] Weak Convergence and Empirical Processes: With …
14 Mar 1996 · This chapter discusses Convergence: Weak, Almost Uniform, and in Probability, which focuses on the part of Convergence of the Donsker Property which is concerned with Uniformity and Metrization. 1.1. Introduction.- 1.2. Outer Integrals and Measurable Majorants.- 1.3. Weak Convergence.- 1.4. Product Spaces.- 1.5. Spaces of Bounded Functions.- 1.6. Spaces of …
Weak Convergence and Empirical Processes - Google Books
12 Jul 2023 · This book provides an account of weak convergence theory, empirical processes, and their application to a wide variety of problems in statistics. The first part of the book presents a thorough treatment of stochastic convergence in its various forms. Part 2 brings together the theory of empirical processes in a form accessible to statisticians and probabilists.
Weak convergence and empirical processes - GitLab
Weak convergence and empirical processes Soumendu Sundar Mukherjee Indian Statistical Institute, Kolkata April 6, 2021 Warning: These course notes (for M.Stat. second year students) have not been sub- ject to very careful scrutiny, and so there may be various typos/mistakes here and
A. W. van der Vaart Jon A. Wellner Weak Convergence and Empirical Processes
The weak convergence theory developed in Part 1 is important for this, simply because the empirical processes studied in Part 2, Empirical Processes, are naturally viewed as taking values in non- separable Banach spaces, even in the most elementary …
EMPIRICAL PROCESSES: Theory and Applications - University of …
2. WEAK CONVERGENCE: THE FUNDAMENTAL THEOREMS 5 2 Weak convergence: the fundamental theorems Suppose that T is a set, and suppose that X n(t), t∈ T are stochastic processes indexed by the set T; that is, X n(t) : Ω 7→R is a measurable map for each t∈ T and n∈ N. Assume that the processes X n have ‘.},
Weak Convergence - SpringerLink
Weak Convergence and Empirical Processes. Weak Convergence Download book PDF. Aad W. van der Vaart 3 & Jon A. Wellner 4 Part of ...
Weak Convergence and Empirical Processes
The third goal is to illustrate the usefulness of modern weak convergence theory and modern empirical process theory for statistics by a wide variety of applications. Accordingly the book consists of three parts, each of which is self-contained to a certain extent; namely, Part 1, Stochastic Convergence , Part 2, Empirical processes ; Part 3; Statistical Applications .
Weak Convergence and Empirical Processes - Google Books
9 Mar 2013 · The weak convergence theory developed in Part 1 is important for this, simply because the empirical processes studied in Part 2, Empirical Processes, are naturally viewed as taking values in nonseparable Banach spaces, even in the most elementary cases, and …
Empirical Processes: General Weak Convergence Theory
stochastic convergence in van der Vaart and Wellner (1996) which will be referred to sparingly. Our goal here is to develop the extended weak convergence ideas to the extent required for a proper understanding of the characterization of weak convergence of the empirical process in terms of nite-dimensional convergence and asymptotic equicontinuity.
Weak Convergence and Empirical Processes - Springer
About this book. This book provides an account of weak convergence theory, empirical processes, and their application to a wide variety of …
[PDF] Weak Convergence and Empirical Processes: With Ap…
14 Mar 1996 · This chapter discusses Convergence: Weak, Almost Uniform, and in Probability, which focuses on the part of Convergence of the …
Weak Convergence and Empirical Processes - Googl…
12 Jul 2023 · This book provides an account of weak convergence theory, empirical processes, and their application to a wide variety of …
Weak convergence and empirical processes - GitLab
Weak convergence and empirical processes Soumendu Sundar Mukherjee Indian Statistical Institute, Kolkata April 6, 2021 Warning: These …
A. W. van der Vaart Jon A. Wellner Weak Convergence …
The weak convergence theory developed in Part 1 is important for this, simply because the empirical processes studied in Part 2, …
