Weak Convergence and Empirical Processes: With Applications to Statistics

ISBN-13: 9783031290381 / Angielski

A. W. van der Vaart;Jon A. Wellner

Opis

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.

Spis treści

Preface (vii)

Reading Guide (ix)

​Part I: Stochastic Convergence 

1.1 Introduction: (1-6) 

1.2 Outer Integrals and Measurable Majorants: (7-16) 

1.3 Weak Convergence: (17 - 30) 

1.4 Product Spaces: (31-35) 

1.5 Spaces of Bounded Functions: (36 - 44) 

1.6 Spaces of Locally Bounded Functions: (45 - 46) 

1.7 The Ball Sigma-Field and Measurability of Suprema: (47 - 50) 

1.8 Hilbert Spaces: (51 - 53) 

1.9 Convergence: Almost surely and in probability: (54 - 58) 

1.10 Convergence: Weak, Almost Uniform, and in Probabil- ity: (59 - 68) 

1.11 Re_nements: (69 - 72) 

1.12 Uniformity and Metrization: (73 - 76) 

1.13 Skorokhod Space (new): (77 - 106) 

1.14 Notes: (107 - 111)

Part 2: Empirical Processes: (113 - 370) 

2.1 Introduction: (114 - 129) 

2.2 Maximal Inequalities and Covering Numbers: (130 - 151) 

2.3 Symmetrization and Measurability: (152 - 167) 

2.4 Glivenko-Cantelli Theorems: (168 - 174) 

2.5 Donsker Theorems: (175 - 181) 

2.6 Uniform Entropy Numbers: (182 - 206) 

2.7 Entropies of Function Classes (new title): (207 - 238) 

2.8 Uniformity in the Underlying Distribution: (239 - 248) 

2.9 Multiplier Central Limit Theorems: (249 - 262) 

2.10 Permanence of the Glivenko-Cantelli and Donsker Prop- erties: (263 - 279) 

2.11 The Central Limit Theorem for Processes: (280 - 299) 

2.12 Partial Sum Processes: (300 - 306) 

2.13 Other Donsker Classes: (307 - 312) 

2.14 Maximal Inequalities and Tail Bounds: (313 - 348) 

2.15 Concentration (new): (349 - 362) 

2.16 Notes: (363 - 370)

Part 3: Statistical Applications: (371 - 558) 

3.1 Introduction: (372 - 377) 

3.2 M-Estimators: (378 - 403) 

3.3 Z-Estimators: (404 - 415) 

3.4 Rates of Convergence: (416 - 456) 

3.5 Model Selection (new): (457 - 467) 

3.6 Random Sample Size, Poissonization, and Kac Processes: (468 - 473) 

3.7 Bootstrap: (474 - 488) 3.8 Two-Sample Problem: (489 - 495) 

3.9 Independence Empirical Processes: (496 - 500) 

3.10 Delta Method: (501 - 532)) 3.11 Contiguity: (533 - 543) 

3.12 Convolution and Minimax Theorems: (544 - 554) 

3.13 Random Empirical Processes: (555 - 572) 

3.14 Notes: (573 - 579) 

Appendix: (581 - 623) 

A.1 Inequalities: (582 - 589) 

A.2 Gaussian Processes: (590 - 605) 

A.3 Rademacher Processes: (606 - 607) 

A.4 Isoperimetric Inequalities for Product Measures: (608 - 612)) 

A.5 Some Limit Theorems: (613 - 615) 

A.6 More Inequalities: (616 - 621) 

Notes: (622 - 623)

References (637)

 Author Index (665)

Subject Index (669)

List of Symbols (676)

O autorze

A.W. van der Vaart is a Professor of Statistics at Delft University, the Netherlands. He earned his Ph.D. in Mathematics from Leiden University. His research interests are in statistics and probability, as mathematical disciplines and in their applications to other sciences, with an emphasis on statistical models with large parameter spaces. He is a member of the Royal Netherlands Academy of Arts and Sciences and recipient of the Spinoza prize. He is a former president of the Netherlands Society for Statistics and Operations Research and served the national and international mathematical and statistical communities in various capacities. He has authored or co-authored eight books, one awarded with the DeGroot prize.

Jon A. Wellner is a Professor of Statistics at the University of Washington, Seattle. He earned his Ph.D. in Statistics from the University of Washington. His research interests include uses of large sample theory in statistics, theory of empirical processes and probability in high-dimensional settings, and efficient estimation for semiparametric models. He is also interested in statistical methods under shape constraints. He is a member of the American Association for the Advancement of Science, the Institute of Mathematical Statistics, the Bernoulli Society, and the International Statistical Institute, as well as the Mathematical Association of America, the Society for Industrial and Applied Mathematics, and the American Mathematical Society. He is a past President of the Institute of Mathematical Statistics, has served as an editor or co-editor of the Annals of Statistics and Statistical Science, and has co-authored or co-edited ten books.